I recently started our aquarium. It's a 20 gallon and right now we only have 4 guppies; because we know we need to cycle the tank. This is basically setting up the biological filter. There are a couple ways to do it; including frequent water changes, or using bacteria from another established filter. If you don't have a filter (which eventually you can get balanced with the right bacteria) you'll need to regularly change water. But of course you want either bottled or treated water to remove the chlorine. Simply put, without thebiological filter the fish eventually die due to the toxic ammonia and nitrites caused by their waste. With the right bacteria established, the environment balances out. I'm glad someone pointed this out to me when starting our tank or I'd probably end up with a lot of dead fish.. =( most people think they can buy the filter & pump and be set. But that's not the case... cycling the tank is the most important thing in keeping fish alive!
Tips for Cycling Your New Aquarium - The First Tank Guide - Getting Your Fish Tank Up and Running with Minimal Headaches
All that said, bettas and goldfish have very different care requirements. I currently have my 20gal setup that just has guppies for now; but I also have a separate betta bowl. For the smaller bowls that don't have filters, regular water changes are a must.
Also, golfish are coldwater fish. Whereas a betta prefers a warmer climate. That said, though, putting a heater in a 3 gallon tank can actually be dangerous.
Here's a good link to some info about betta housing (and the Home page has everything else you'd ever want to know!
My last betta lived 6 years, I really loved that little guy!!