Awakening has 4 difficulty modes- Normal, Hard, Lunatic, and Lunatic+. Each one can be played on Casual (you can save mid-battle and dead units return at the end of each map) or Classic (what you played). To unlock Lunatic+, you need to beat Lunatic, but if you only beat Lunatic Casual you'll only unlock Lunatic+ on Casual.
That said, Hard mode is a small increase from Normal, Lunatic is a huge increase from Hard, and Lunatic+ is completely unfair. To be perfectly honest, playing them makes it feel like Normal is actually Easy, Hard is actually Normal, and the game skips Hard and just throws Lunatic at you. The main differences between Normal and Hard are that enemies have a little more attack and HP, have forged weapons later in the game, and the big one: reinforcements move as soon as they appear, without giving you a turn to react to them. Your units also grow a little slower after reclassing several times, but there are a few more enemies so the EXP mostly balances out.
So basically Hard is your "medium higher difficulty". You'll be fine as long as you don't throw your money away, don't try to train too many units, keep your important ones alive, and recruit a few of the children.
Some tips for FE8:
Bosses (and certain enemies in cht.18) give crazy amounts of exp in this game, so much that you can just give Seth every boss kill and he'll stay up with the rest of your team (despite gaining practically no exp from scrubs), ready to do anything when you need help (and his help is pretty incredible).
After a few chapters, the game forks, and you get to choose whether to accompany Eirika (the starting lord) or Ephraim (playable in cht.5x and 8). After a few more chapters, they join up and you get to use both of them. Unlike Awakening, Sacred Stones has plot promotions (your lords can promote after a certain point in the story). These promotion points occur shortly after the lords rejoin. Because there are no Second Seals in Sacred Stones, your units will have the highest stats if you get them to Lv.20 before promoting them. Thus, in the first few chapters you should focus on packing exp into whatever lord you aren't going with, so they can promote immediately (assuming you want to use them both later, which is advisable because they have very strong prf weapons.
Thieves have the ability to steal held items in this game (items, not weapons). As long as they're faster than their target, you can walk up and use the steal command to grab stuff. Usually, this will get you vulneraries (which you can sell for cash), but occasionally there are gems (bullion) and two bosses even have powerful defensive items that you can steal to weaken them. But be careful, because enemy thieves can and will steal from you.
Manaketes are weak to arrows in this game. Keep that in mind, or you could get mauled.