Every fight is three against three: your character and two creatures against the enemy's. Each round every creature picks a move, then the moves play out in speed order.
1Your side and theirs
Your character and your two party creatures stand on the left (1) with their level and HP; the enemy team mirrors you on the right (2). The glowing slot is the one choosing a move now.
2Pick a move
The parchment lists the acting creature's four moves (1) with their power, hit count and effects; click one to choose it.
3The ultimate
Your class ultimate (1) is a once-per-battle action that goes before everything else; click it when it is lit.
4Aim at a target
A move that needs a target waits for you to click an enemy (1); the ones it can hit are outlined.
5Lock in the round
Once every creature has a move, LOCK IN MOVES (1) plays the round. SPEED ORDER (2) shows who acts first; the faster creature strikes before the slower one.
6Autoplay, forfeit and the log
AUTOPLAY (1) fights for you and keeps climbing; its cog opens the autoplay settings. FORFEIT (2) leaves the fight - it counts as a loss. COMBAT LOG (3) replays what happened this battle.
7The streak
Every win on an uncleared floor adds to that floor's streak; the ASCEND portal (1) counts down the wins still needed. A lost battle resets the count to zero - you stay on the floor and rebuild it. Full HP comes back after every fight.
Screenshots are taken from the live game by tools/tutorial_shots.py (dfmc-client) and regenerated after interface changes. The rules behind each screen: system-stage, system-streak.