Protip: if you die in the Mines and something you absolutely need (like your primary weapon) is lost, quit out, abandoning that day unsaved. Reload and restart that day. It's not worth losing something that isn't super easy to replace (especially early on, they can be quite expensive for your income at that point). Later on, other weapons become even more expensive.
And yes, the Adventurer's Guild is where you want to go. My recommendation is the Bone Sword. It's super quick, a sword's wide swing, and the best damage of the things that should be for sale there, if that's how far you've gone in. It'll run you 6k g.
There you can also check the list on the north wall, that's the number of monsters you've killed and how many you need to kill to get rewards from Gil, the guy sitting on the right. Some of those are REALLY nice. (My favorite is the reward you get for killing slimes.)