Aside from dish soap, which I've heard can cause problems to the rubber on your tires over time, I just picked up a gallon of rubber lubricant from Napa. I put it in a spray bottle and just spray some on whatever part of the tire I'm working off.
I don't know how true it is that dish soap can be bad for your tires, but I've heard something about it eating away at it because of the petroleum and something else and something else because dish soap is meant to cut through grease/oil. I know you can rinse it off, but some will still be stuck on the tire inside the rim where you can't reach it.
Balancing tires is super easy. The first time I just used a $4 metal rod from home depot, but recently I picked up a balancing stand from HF when it was on sale for $40 which makes things pretty easy. Just spin the wheel a couple times and stick the weight on the spot that always stops at the top. Once a different spot stops at the top each time you're done.