You can do 50% once a week water changes in your 3 gallon. Leave the fish in, and just be gentle and watch out in case he's curious and wants to see what your gravel vac is lol
And a correction to Sena's post -
Originally Posted by Sena Hansler
The tank is running with a filter on, and the use of something raw, or even just fish food will kick in ammonia. For 4-6 weeks this tank runs without fish. This keeps ammonia, nitrite and nitrate in the "cycle" where levels are low and harmless. Ammonia should only spike if the tank is uncleaned, or adding another fish (not recommended with bettas in stuff under I would say 7 gallons... personally. except maybe a snail...)
The levels of ammonia, and nitrite are not
safe while the tank is cycling, even when you are doing a fish-less cycle. Ammonia will spike at the beginning of a cycle, and this is normal. If you keep your fish in during the cycle, you will need to do 50% water changes every day, and the use of Prime which detoxifies ammonia and nitrite is also helpful.
You can keep an uncycled/unfiltered tank/bowl. The frequency of water changes will depend on the containers size
eta: For the first month or so while your tank is cycling, 50% water changes daily, and use a product like Prime or NovAqua+ to detoxify the ammonia and nitrites. Once you have some bio bacteria in your tank to handle the load from your one fish, you can change 50% of the water once a week. I have had a 2.5 gallon and a 3 gallon cycle. You just have to watch it carefully - the best way to do this is to buy a test kit. The best one to get is the API master test kit. You'll be able to watch your ammonia, nitrite, nitrates and pH with it. Don't get test strips, they are basically useless.
Congrats on your new betta! Cared for properly (heated, clean water and a variety of quality foods) he will give you years of entertainment and joy :)