I'm tight on money, but I would like to try something like Anvil of Dawn or Ultima Underworld. They seem to be awesome games, but I don't have the money over to just try them.

Do you know games, which have a demo I can try or a free to play version?

First-person dungeon crawling?
There's a free clone/port of Dungeon Master http://dmweb.free.fr/?q=node/851

For single character crawling there are the old Elder Scrolls: Arena and Daggerfall, complete with easy to install packs.
http://theelderscrolls.wiwiland.net/?title=Arena_:_ArenaSetup_EN
http://theelderscrolls.wiwiland.net/?title=Daggerfall_:_DaggerfallSetup_EN

System Shock and Arx Fatalis are both close relatives of Ultima Underworld. I don't know where you can find demos, but Arx is available here on GOG.

That was fast! Thank you :D I played Arx a lot, when I was younger. That game was really good. I baked apple pie all day in it xD