Wood Helmet

You can wear the full set before you have found a single ore vein, and that is the whole argument for it. Chop, build a workbench, put it on. The protection is small enough that it will not save you from anything that was going to kill you, but early slimes chip you less, and the wood was coming out of the trees anyway.

Wood Greaves are the only armor in the game that do nothing whatsoever on their own. No protection, no effect. Two other pieces also give no defense, but each hands you something else in exchange. If you are wearing a partial set, the legs are the piece to leave for last, and there is no reason to keep them equipped once you have moved on.

Almost at once. The nearest desert gives you cactus armor for the price of a walk and that beats the full wood set outright, and the first copper or tin from a cave does the same. Sets made from corruption or crimson trees are a small step up if you happen to be standing in one, but none of the wood tiers hold up past the first day.