If you've got iron armor, you can take out the Eye Of Cthulhu and Eater Of Worlds bosses easily. Skeletron (the boss guarding the dungeon, sometimes called castle), is requires a decent weapon to beat, such as a flail, a minishark, or anything in the middle.
You can go to the corruption area and break shadow orbs, go to the jungle looking for some nice gear (collecting stingers/vines/roses and keeping them, they are very useful!), maybe a meteor or two will fall. Once you've beaten Skeletron you can venture in to the dungeon. Once you've got full Shadow, Necro, or gold armor you can take on the Underworld fairly easily if you know what you're doing.