I finally found myself enough confidence to put together a Tier List for this game. I'll try to give a good explanation for my decisions here, and of course if someone is higher on the list, I believe the unit will be more useful in long term. Currently I have a finished BL and GD run, and doing pre TS BE run.
Byleth - As a main character, with his personal skill he becomes one of the best early game units for you, plus you need to deploy him in every story chapters - which is not a bad thing. Byleth has one of the best bases and groths out of the gate, which only gets better once he gets his personal weapon and unique class. For that, training him is easy, and it's best to train his in swords, unless you want to raise other skills for recruitment.
Lysithea - When it comes to stats, Lysithea is really minmaxed, but what she for what stats needs - already has good growth. She's fragile though, so only use her for her magic from a safe distance. Getting Thrysus for her is advised, and luckily she starts in the same class as Lorenz, who has the paralogue to aquire that. Lys also learns spells much sooner than other units, and her arsenal is just amazing: Seraphim, Dark Spikes, Luna, Warp. Gremory path is the best for her, but if she has the time to train, she can be also a Dark Knight.
Catherine - Cat joins as a Swordmaster at Lv15, has the Thunderbrand from the start, and a very solid offensive unit in a long time. She's fast enough, can learn Healing Focus with little effort, and one of the units who doesn't even need a class change at all. I believe if she ever falls behind, she wouldn't have a difficulty to catch up with the other units. For enemies she can't kill instantly, she can handle them with Levin Sword most likely.
Petra - Petra is fast, probably the fastest unit in the game. She also has great Skill proficiencies, so you can turn her into a Swordmaster or a Wyvern Rider, depending on your style. With her personal skill she can get the highest crit rate in the game, which is most likely an instant kill for units with less then 50% health remaining.
All the lords - I want to be fair and say, all the lords are amazing. Claude is a great flying archer without you need to worry about his skills, Dimitri is a strong and tanky lance unit, while Edelgard has a lot of potential with Axes and Dark magic. You need to use them a lot, so training them would be just beneficial for you, and you are not going to regret it.
Ingrid - Ingrid is just meant to be a flying unit, and not a bad one either. As a Pegasus rider, she comes with high speed and very good resistance - later turning her into a very dodgy flier most likely. Since she's going to keep training in Flying, Alert Stance+ will be unlocked easily, adding that up with flier Avo +10 bonus, Evasion Ring and a batallion, Ingrid can be one of the best dodgetanks in this game, who can also double her enemies. She also has a nice set of spells, but unfortunately fliers can't use magic.
Sylvain - Sylvain can be instantly recruited if you are female Byleth, and he's best used as a cavalier unit - having proficiency in lances and horses as well. He can decently endure battles, and he gets even better when a female unit is next to him. At A lances he unlocks Swift Strikes, which is a must for him. Easy to train into a Dark Knight with his budding talent, making him a versatile unit. He has one of the earliest paralogues that can grant you a crest weapon.
Mercedes - Easily the best healer in the game. Mercedes can learn not only Physic, but also Fortify. Thanks to her crest, she can also conserve uses of recovery magic. Her personal ability is one of those that can contribute even in the lategame, a 3H version of Live to Serve. She has a budding talent in bows, which is a nice utility, but she's better off learning some reason magics instead of that. If you could, try to get her the Rafail Gem.
Felix - Felix is another strong sword user, and he probably should stick swords only. His budding talent is a bid odd, probably useful if you want to turn him into a Mortal Savant. He is also the only unit that is better when deployed without a batallion. His crest weapon can only be obtained in BL route though, and relic reward from his paralogue can also be missed if you are not saving enough people.
Marianne - She would be the best choice for the dancer class. Not only making her a much better support unit, but it is also a class that can use magic, and benefits from using a sword. From her paralogue she can get Blutgang, which is a great crest weapon for her. Her offense might not be the greatest, but either then she can trust in Faith magic. Has proficiency in flying too, so has a potential to gain Alert Stance.
Seteth - Starts with high Lance and Authority, not to mention as a Wyvern Rider. He can get Spear of Assal from his paralogue, which kills most cavalry units in his way. He can also learn Swift Strikes, and he can also prevent counterattacks sometimes. A solid choice if you need a flier unit.
Hilda - Fast and strong, but her accuracy might be a problem for some time. She can also get her crest weapon Freikugel very early, which comes with a strong Combat Art. While she can learn Bolting in Reason, it's not really worth it, and her Budding Talent is also not needed. If you can keep her accuracy good enough, she can be rewarding. Turning her into a Wyvern would be the best path.
Ignatz - Archers are having a great time in this game, and Ignatz is starting out pretty well. Having a +20 Hit on every attack, he's guaranteed to get some hits and kills. He also has very good growths in speed, luck and dexterity. He can be a great support unit too with Break Shot, Ward Arrow, and even Seal Strength. Better trained on the Sniper -> Bow Knight route, Ignatz will appretiate the extra range these route give to him.
Leonie - Leonie is another potential bow knight, but she's better off starting as a cavalier instead as a sniper. She also has great dexterity and speed, not to mention a nice defense too. Overall a solid unit, just use one of them as a bow knight instead of the others - if one of them gets screwed in their growths that is.
Ferdinand - Ferdinand is designed to be a cavalry unit as well, just like Sylvain. He might struggle early on, but he has a potential to become a very fast and dodgy unit. His personal ability grants Hit/Avo +15 when unit is at full HP, which is not a tall order to maintain, especially if he gets the crest weapon that also heals him at start of his turn. He can also learn Swift Strikes, and I'd train him in flying as well just to unlock that nice Alert Stance+ for him in the long run. His budding talent might not worth it though.
Annette - She is a weird unit when it comes to proficiencies. Annette prefers Reason, Authority and... Axes. Annette can be a really powerful mage, and with C axes she can also unlock Lightning Axe - a very powerful combat art that could one shot enemies, even when she has just a Training Axe. She can get later the Crusher, which has a great combat art with it for her. Unfortunately she's a very squishy unit, and she might have a hard time surviving her own paralogue.
Hubert - Also one of the mages that have problem with survival, but he's one of the 3 mages who can wield dark magic without the dark classes. Very good in early game, but unfortunately he's only available in Edelgard's route.
Dorothea - She would be my 2nd choice to be a dancer. Has a nice set of reason magic, and a personal ability that can heal adjacent allies. Has good magic and resistance, but low strength and defense.
Linhardt - Black Eagles healer, who can quickly unlock Physic, and one of the few people who can actually use Warp. Linhardt can also heal himself with his personal ability, but it only works if he does not take any action that turn. His magic growth is ok, so he can be a good warp mage later in the game.
Dedue - Dedue has great growths in Hp, Str and defense, while lacking in speed and resistance. Good to be a Fortress Knight, and he gets extra defense when he doesn't take an action - making him a great bait unit. His availability is not great though, he joins later post timeskip, and he is exclusive to BL route. His biggest weakness is mages who double him. Armored Strike is probably his best skill.
Raphael - Raphael is another unit that is great as a Fortress Knight/Great Knight. His Spd growth is non-existant, but when he starts in brawling and doubles the enemies before they hit him, he doesn't need speed. He's very bulky, and hardly dies in the GD route, walling off brawlers and monsters easily. His personal skill is not very good though, and Great Knight can only use gauntlets when dismounted, so he might need to switch to axes later on.
Flayn - Flayn has a very high resistance, good magic and learns Rescue and Fortify easily. Weirdly she also has proficiency in lances, so she might take the path to become a Holy Knight - a Rescue user with canto. Her offense might be problematic early on, but she still worth the investment.
Bernadetta - Bernadetta's only good skill proficiency is in bows, which could be good with her speed and dexterity growths, but she might suffer in strength growths. If however she gets out of it, she can be dangerous. She has the budding talent Pass, allowing her to pass through enemy tiles. Her crest also gives a chance to attack twice, which is basically a brave effect on one attack. Bernadetta has a personal skill that gives her an attack boost as well, but all these are a bit situational for my taste. Still, Deadeye and Encloser are nice skills to have.
Shamir - Shamir starts as a great bow user as well, but she also has the chance to get screwed in growths, and her skills are not much outstanding compared to others. Her personal ability is very good though, when initiates combat and defeats foe, grants Str/Mag/Dex/Spd +4 for one turn.
Manuela - Manuela has good starting strength and magic, but her highest stats might be speed and charm. She can learn Bolting and Warp once reached A in Reason and Faith. She is in the same category as Marianne and Dorothea, but she cannot be turned into a dancer, and her personal ability is also not very useful. 35% growth in both Str and Mag is also not very strong, but if her stats turn out good, I'd recommend deploying her.
Caspar - To be fair I still need to experiment with Caspar, but he looks like a good brawler unit on paper. His Strength is pretty good, gauntlet skills look very interesting,
Ashe - Ashe is probably best as another archer, where he can unlock Deadeye. But other than that, his abilities are nothing outstanding. His personal ability is unique for him, but thanks to the always open shop and the high amount of gold you'll get, keys will be your smallest problems in this game. He can get good dexterity and speed, but everywhere else he might fall behind.
Lorenz - Lorenz is slow, and that's his biggest problem. He could be a strong mage, but not strong enough to kill his enemies in a single hit. It's tempting to train him as a cavalry unit, and wihle Frozen Lance is very nice, it'd just leave him in a spot where enemies will double him. He can become somewhat useful once he become a mounted mage knight, but besides getting the Thrysus, I'd not recommend using him.
Gilbert - He's really not that great. Good for Annette's paralogue, but he only appears in Dimitri's path - where Dimitri and Dedue can easily replace him. Like his as a character, but not as a unit.
Alois - Alois is another odd unit. Looks like he should another one of the armor knights, but somewhat an oddball amongst them. His gauntlet skills look interesting, but his axe skills are not that beneficial - exhaustive strike just doesn't worth it.
Hanneman - Hanneman looks like a good mage unit when he joins, but he really need some training. Not sure if it's worth it though, his bow skills are bad, and his magic is only good if he one shots his enemies. That's right, Hanneman is very slow and very fragile, and very one dimensional. If he turns out to be something, he can unlock Meteor, but that requires A+ is Reason skills.
Cyril - People say he can be turned into a good Wyvern Lord, but in this game that could be true to most of the units. His bases are bad and his growths are bad, so instead of having a useful personal ability, he gets an ability to fix up those growth rates. I believe his stats are calculated without the personal skill bonus, so you've got some work ahead of you to make him a good unit, and it feels like a big gamble, as his combat arts are also not very good for him.