Before the FR gauge is filled, you will probably start spamming a few Curagas. When Overdrive reaches 5, you can press EX and LD to get the most of his damage. After his first set of upgraded EX & LD, it will take less Curagas to reach Overdrive 5 until eventually the FR gauge is full. If you use Braska's FR, he can capitalize very well with his upgraded EX and/or LD in these turns should he act; otherwise another fire attacker (or Non-elemental but fire-enchanted attacker) can use a strong BT phase to deal good damage.
Curaga (S1) is his battery/healing button. Can be used for it's utility, FR gauge filling focus or to buy turns until his Overdrive reaches 5 stacks.
Ultima (S2) is a filler skill if you're running low on Curaga's to spam
Ultimate Jecht Shot (EX) should be used when it's upgraded (Overdrive at 5 stacks) for the biggest OOMF
Jecht Bomber (LD) can be used as a big nuke button, but since his LD buff is tied to it be mindful to not overuse it in long endurance fights
Mixture Power (FR) has decent on-turn conditions which Braska himself can usually hit, however other units may need help via enchanting (Ignis can enchant non-elemental attackers while Lulu and Neon can enchant anyone). In HP Poison/Attack heavy fights, units with strong overheals and regens can help ensure units are healthy enough to hit his full HP FR condition.
Final Jecht Shot (BT) finally gives Braska a party fire enchant and a cool follow up after his actions (including Curaga). This bumps up his damage quite a bit and allows allies to fulfill his FR.