How does the dungeon work even? I've made a few runs before, but I'll end up in a save room and suddenly restart back at floor 1 again.
Advancing through floors is fairly straightforward, just exit a room to advance a floor. The number at the start of each room shows which floor you're on.
Save rooms show up semi-randomly while exploring. Every time you save your game you'll reduce your floor count by half (e.g. saving on floor 50 puts you back to floor 25). This includes if you make multiple saves in the same save room; you'll halve your floor count every time you press confirm at the save point. You can use the pink crystal to change weapons etc. without penalty.
Exiting and re-entering the dungeon (by talking to the octopirate and choosing to leave/enter) resets your floor count to 1 every time.
There're 4 different save rooms, including the one you start in. 3 of them will show up while you're exploring without needing any special requirements. You should talk to the octopirate at least once in each different save room, it's a requirement for unlocking the 4th save room and the bonus boss. Also, you apparently can't advance past a certain floor count unless you speak to her in each room. You should keep talking to her until she stops saying anything and you only get the option to exit.
After you've talked to octopirate fully in all 3 save rooms, from then onwards you have a small chance to find the 4th save room when exiting a room under certain conditions. We're not exactly sure what those conditions are, but it only seems to happen in the cave area or cave save room, and only when you've explored a lot of the dungeon already. It may be related to your current floor count, it may be related to the second number you see when talking to octopirate (which increases the more you explore and doesn't reset when you leave the dungeon, though it can decrease).
The 4th save room kinda looks like an abandoned lab and has more exits than any of the others. The top left exit takes you to the NPC gallery, the bottom left one takes you to the bonus boss.
I'm pretty sure that once I went straight to the 4th save room as soon as I left the dungeon's starting room (i.e. when I was on floor 1), but I'd already found it once by that point so the rules for accessing it might've changed.
Those're the main points. There's some other stuff like the stats on dropped weapons improving the further in you get, and the direction you exit a room affecting the next room, but you'll figure that out yourself.