You can go exploring some corrupted/crimson areas (randomly picked at world generation) to destroy the shadow orbs/crimson heart waiting for you at the bottom of chasms.
Each shadow orb/crimson heart destroyed will provide you some loot, and make a meteorite land on the world. Once the meteorite has fallen, go find the site, and mine the meteorite ore. This will allow you to craft some meteorite gear and weapons.
Once you've destroyed three shadow orbs/crimson heart, it will make the Eater of Worlds/Brain of Cthulhu spawn, allowing you to fight two different bosses.
Note: The shadow orbs are protected behind some ebonstone/crimstone, which you will not be able to mine with your current pickaxe. Just buy some dynamite from the Demolitionist and use it to access the shadow orb. Or you can buy purification powder from the Dryad, which turns the ebonstone/crimstone into the easily destroyable standard stone.
From 1.1 With Love
Spawn Switching
By using a on/off wire trigger block you can effectively "open" your bed's wall and invalidate the room it's in, leading you to spawn back at the middle of your world. If you're close enough (wiring only works for 2000 blocks or so) you can just hook up a switch at your spawn that'll change the blocks again and allow you to spawn back at your home base. Otherwise, you might be able to coax a contraption with a statue spawner/pressure plate/lava block to toggle it for you automatically.
This post on the TerrariaOnline forums in particular has an interesting bird-based reset circuit that seems to be fairly popular, but you'll need a bird spawner.
Digging the hard way
From your main base, just dig a hole straight down (the classic "Hellevator", if you will). When you get tired of exploring your caves, just start digging left/right towards your shaft. It'll take a while, but you'll at least probably find some goodies (more caves, ores...) on the way, and getting back will be a lot easier. You can even use a depth meter to align all your "escape routes" to the same depth, so that eventually you can just have one long path across the world that speeds up travel.
Previously, on Terraria...
A trick that worked in 1.0.5 was to start up a multiplayer server and explore with a friend (or just explore with two characters locally), then follow the instructions:
- Build a room with a bed. A room isn't necessarily required, but will keep the other person safe. It's best to put the bed in a corner.
- Set your spawn point to the bed.
- Break the bed and use a mirror. Logging out might work but I don't make any promises.
- Do whatever you need at your spawn point. When you're done, let your friend know.
- Put the bed back down in the exact same spot. This is important.
- Use the mirror again.
If you did it properly, you should travel back to where the bed is. This might not work in 1.0.6, though, and it's only good for exploring together -- if either of you get separated, you'll have to travel back manually.
If you're exploring solo, your only real option is just to take the time to improve the path. Wooden platforms over minor gaps and flattening out bumps will help a lot with horizontal movement, and creating a double shaft with a soft landing area (e.g. water) on one side and a wooden platform tower on the other will make vertical travel easier. Using background walls other than dirt will also stop the enemies from spawning.
Best Answer
Time in the "base world" actually has to pass for things to grow, corruption/hallow to spread, etc. For anything to happen basically.
So no, plants won't grow if you're off looting someworld else.
An addition/exception by Bob from the comments below: The player doesn't have to be physically in the world; the world just has to be running. One can leave the world running using the dedicated server launcher.