The biggest difference that separates FE4 from other FE games is resource management; every unit has their own gold only they themselves can spend (maximum per unit is 50,000G), nothing can't be traded from one unit to the other*, and units can't buy weapons that they cannot use (however, it is possible for a unit to get a weapon they can't use in their inventory if they kill a unit that drops the weapon). Every castle in a map you own has a pawnbrokery you can sell items to that your other units can then buy for themselves. All items that have durability can be repaired at an armoury (also in every castle) for a price, so don't worry about seemingly having little weapons of staves.
*There are two exceptions to this: two characters who are paired up with each other (you get a pre-determined pairing during the Prologue) can "transfer" gold to and from themselves as long as they're adjacent to one another. They transfer either all their funds to the other, or enough until the recipient's gold cap is met. The other exception is those of the Thief class, who can do this action to anyone regardless of pairings.
Follow-up attacks in this game are only guaranteed if the unit has the Pursuit skill, and the effective/true speed you need to perform follow-up attacks is 1. There are other skills that can allow follow-up attacks, but they're much less reliable to proc.
Units in this game will have a mixture of class and personal skills. It's important to note which skill is what on what unit (for example, Sigurd has Pursuit as a personal skill, but anyone in the Myrmidon class will have Pursuit as a class skill) as, well, there's a village you can visit in Chapter 5 that will allude as to why it's something to note. I won't go into any more detail than that.
Most important tip though: try to have fun. FE4 is one of my favourite FE games large in part because Jugdral is one of the best worlds that exist in a Fire Emblem game--the first half of the game is pretty special.