One thing you may want to consider is keep one of each type of weapon in the bank, and take the time to keep each skill current (or nearly so). Don't buy the uber of uber weapons, you replace weapons and gear so quickly as you level it isn't worth spending a lot. Just troll the AH for decent items that you think are affordable, or simply wait for them to drop.
Skilling up Melee will take time - you will level far faster just letting the pet tank and DPSing from range - but there will be times when you just have to get up close and personal. These situations should be avoided, but it is nice to be ready for them when they arise.
You never know what weapon type will drop next for your toon. Best to be ready for anything. So, to answer the OP's question, the best melee weapon for you is the one with the best stats, whatever its type is. Just be ready for it. It would be a shame if you found a really nice 2H Axe, and found your skill in 2H Axe so low that you can never hit anything with it. Your melee weapon isn't decoration, it should be functional too.
Get a good gun and a crossbow (if you can find one - you can skip crossbows if you want) as well. Keep all ranged skills up to date. As I leveled up, it varied which was better for me at any given time.
Shelor007