Rotating the tires is no problem- all you need is a jack and a pair of jackstands.
Balancing the tires needs to be done professionally. There are gizmos you can buy that lets you do the job yourself, but they involve balancing the wheel on a central pivot and adding weights as needed. A pro will use a far more accurate dynamic balancing machine that will spin the wheel and tell exactly where and how much weight to add. Normally a new car's tires are already balanced, but you'll still need to have it done every so often due to tire wear, lost weights, etc.
The next time you buy tires, spend the extra $10 per tire for lifetime rotate/balance. It's the best money you'll ever spend, since it's a job really not worth doing yourself.