As another big tip, don't get into combat, avoid enemies as much as possible. This goes double for the grubs that will spawn into flies in the upper levels, and also particularly for slimes that have a lot of health for not a lot of return.
If you find yourself getting into trouble in the cold section (40-80) and you haven't lucked into something like the Forest Sword or the Wooden Club yet, consider the Wooden Mallet for 2k at the Adventurer's Guild once you hit level 40 in the mines. It's relatively hard-hitting, swings surprisingly fast, and the right-click can be used as a 'get away from me' panic button. There's also something of an exploit concerning the hammer class weapons involving using the right click ability then spamming left click to hit multiple times. But even without that, the Mallet is a solid weapon for 2k, and should easily carry you to the Lava Katana on level 90.
Failing that, if you prefer a sword, the Claymore unlocks at level 45 and is also 2k in price. It swings slower, but does more damage per hit. Sadly, the alleged bonus to defense doesn't actually function due to a bug in how the code is handled.
Either way, the Steel Smallsword you get at level 20 should at least carry you to 40-45 and that weapon should carry you to the Lava Katana if you are careful. If you get lucky finding a club or the forest sword or some other weapon drop (Neptune Glaive?) then you're pretty good to go. But generally, avoid combat and you should be fine.