Will be traveling to Bangkok then taking a roundtrip flight to Phuket, which airport is cheaper/easier to fly?

It doesn't matter, both airports are roughly the same distance from cental Bangkok and neither one is necessarily more convenient than the other. It will depend on the airline you take, Bangkok Airways and Air Asia fly out of Suvarnabhumi, Nok Air, 1,2 Go and most other domestic flights leave from Don Mueang, Thai Airways flies in and out of both airports. If you have an international connection, Suvarnabhumi will be much more convenient. Be careful of low baggage allowances and high overweight charges on the budget airlines, and although more expensive, I like Thai and Bangkok Airways the best and much prefer Nok Air over Air Asia. Have fun.