Greetings and welcome to Fish Forum.
Jersey gal' eh? Lemme' put on my southern drawl for ya'll now and this oughta' be a right good ol' time.
....I kid, I kid.
Yes, you still need to do water changes with a filter if your tank is cycled. You just do them less often. A cycled tank is generally something around 25-50% once a week on water changes, where as an uncycled tank is somewhere more along the lines of 50% twice a week.
Ideal tank size? As big as you can get xD I don't think a fish is going to object to more room to move in. But a good size for a single betta with no other tank mates if you intend upon cycling the tank is 5-10 gallons. Any smaller than 5 gallons and your tank probably won't be able to hold a cycle.
Spring water and you: Hmm, I've heard some horror stories from some water operators about the water system in some parts of Jersey..... were I you I would go to your local water municipality and get a list from them on their read-outs on what all the water contains, and see how your pH, ammonia and heavy metals are reading. Then I'd go get a sample of your tap water and take it to your local fish store and ask them for a test. Actually, no, I'd take a sample of the bottled spring water AND a sample of your tap water, and ask them to test both. Many pet stores will do this for free. Find out which is the lesser of the two evils, and go with that. Honestly though, unless the tap water is in dangerous levels for your fish, I'd say tap water and dechlorinator is your best bet, it's cheaper. Do your research, many places have water that's just fine... some places... I wouldn't swim in that water, let alone drink it.
If you don't already have one, and plan on cycling your tank, go ahead and get a water test kit, the API liquid test kit is a good one I hear. =]
On the topic of making a betta happy - clean, warm water, a big tank to move around in, and decorations (like plants and caves) to explore and a good, healthy diet are generally considered the best ways to maintain a happy, active betta.