Make plenty of disarming tools, and disarm all the dart traps in every room before you start to chisel or take items. If you wish to journey to the ruins for valuable relics and rare gems, this method is for you.
Look out for pressure plates, and only use disarming tools unless absolutely necessary. Killing enemy mobs is not necessary, although you may want to if they are bothering you. Remember to look out though, as dart traps, scorpions, and spikes can easily put an end to you. If this is what you wish to accomplish, simply explore as many rooms as you can until you find the other exit.
If you do not wish for loot from the ruins and simply wish to quickly travel to another island, you way want to ignore the loot and simply get through as fast as you can. Bring some relatively cheap items with you as well, such as twigs or rocks, for weighting down pressure plates. If you have enough oincs, you could replace the cork candle hat with a miner hat, which can be bought from Pig and Piglet's General Store for 20 oincs.
With your resources in hand, proceed to make the following items: A machete, for hacking through vines, a spear or halberd, to defend yourself, a ball pein hammer, for looting relics and gold, a few disarming tools, for disarming traps, some armour, if you wish for protection from scorpions, snakes, and batilisks, and a cork candle hat, to provide light while keeping your hands free. Collect some oincs and kill some guard pigs if you want some better items (explained in the next section). Gather some cork from Claw Palm Trees, flint from boulders and stalagmites, and twigs and grass from saplings and tall grass. Firstly, source some iron from the Painted biome by finding it on the ground, looting Thunderbird nests, and destroying Gnat Mounds. Before you even begin to explore the ruins, you will need adequate tools to do so.