Firstly the surface as much as you can. You can get some handy stuff from chest.plus smash any pots you see for torches, ropes and potions.
Next, underground beneath spawn is a good place. You should be looking for ores, crystal hearts and chests.
As you build up some weapons, hearts and armour, your next best area is probably underground snow. There's some useful stuff down there.
Don't go too deep to start with. Once you feel comfortable and get your bearings you can start going deeper.
I don't want to say too much more, so I don't spoil it for you. If you don't care about spoilers there is a good guide on the wiki that gives an outline of when is a good place to explore each biome.