I cut out the first group of fights before this against the other enemies, since I didn't feel like it had a real point to them.
Boss fight against Elly. I don't know how, but you see this coming from the second you meet her earlier in the game.
While this is the first video you see of Elly, Fei meets her earlier in a forest at the beginning of the game. While they are supposed to be enemies and Elly acts like it, Fei's kindness changes her attitude towards him as an enemy, where Citan tells her to leave while he's sleeping to avoid problems.
- Video spoilers*
The fight against Elly shows that Elly does not actually want to fight Fei, but her use of Drive changes her personality to the point of becoming fully hostile towards Fei. The game never really goes into real detail about what Drive exactly is or does, and this is really the only time you see it used in a fight. But as the dialog mentions, it doesn't give someone power, only the power they have in themselves deep down are brought out. So similar to Fei, there is more to Elly than you currently see...which becomes hilariously clear way later in the game, but at this point in the game you won't think anything of it.
Fei: LV24 - Metal Vest, Leather Hat, Power Ring
This game has two main battle systems. One is regular combat, the other is Gear battles. If you like regular combat...then you'll be a tad disappointed, because the game focuses mostly on Gear combat, and the regular combat system becomes nearly useless at end game. Anyway...
If for some crazy reason you've been neglecting Gear upgrades, this is going to be rather difficult. I almost bricked myself here. There's a FRIENDLY, yes FRIENDLY Gear Repair Robot before this fight. If you're stupid like me and choose to fight it, it'll be destroyed and you won't be able to heal your Gear. At this point in the game, there is no way to heal your Frame HP of your Gear, so if you're low on HP from random fights and killed this guy AND saved...then you're probably stuck.
Anyway, like I said, Gear upgrades are super important in this game. They are expensive, but this game likes to throw money at you a lot sometimes. Sadly there's no way to know which Gear you're forced to use in the story all the time, so sometimes you might upgrade everyones' Gear (yea there are more than one in this game) even if you don't use them. Use a FAQ if you don't want to blow money.
Back to the fight. The Gear battle system is a little more complex, but still pretty straight forward. You have Attack Levels. Every time you attack, it goes from 0 all the way up until 3 (and Infinity Mode late game), although you can't get up to LV2 or LV3 unless you learn more regular Deathblows for characters. If your'e at LV1 or higher, you can perform a Deathblow. In this game, 90% of the time you're just going to want to do T T or T S. (T = Triangle, S = Square). The damage output for LV2 Deathblows aren't usually worth the time it takes to get waste turns getting to it compared to simple damage from LV1 Deathblows that you can do in 2 moves (LV2 would take 3 turns, and LV3 would take 4 turns). As for building up that Attack Level, just use regular attacks. You might as well use X all the time unless you're really low on Fuel, because X will do the most damage.
So now that you know how to dish out tons of damage with your Gear, your strategy for these fights is to just spam that. Yea, really straight forward for so early in the game. If you don't do this, you're in for hell. Take out the groups as fast as possible. You MIGHT want to Booster (this is just like a Haste spell in FF games), but it makes you use up more Fuel per turn. Use at own risk of running out of Fuel if you don't take out everything fast enough.
Finally, Elly time. Do the same stuff. A lot of this fight is dialog based. I don't know for sure how fixed it is, but just to be safe, keep slamming on her as if she has 9999 HP. If you're getting low on Fuel, turn off Booster. Charge restores Fuel, but wasting turns against Elly when you're probably low on HP by now isn't a great idea. Just keep pounding on her and the fight will end hopefully.
Kind of a tricky fight if you're not used to Gear fights yet. Don't underestimate how broken Booster is, it's practically two turns in one, that's how fast it charges your attack bar.