How did you acclimate him to the new water? You may have shocked him. Also the flow might be too much for him. Is it baffled and prefiltered?
To do a water change you should use the thermometer (should be a glass one that goes in the water not sticks on the outside) to make the water coming out of the tap to be the exact same temp. You should then fill up jugs/bucket or whatever you do, then add conditioner, mix it up a bit then add it to your tank before it has a chance to cool. I also cup my boys in opaque plastic solo cups while the water is being changed, and then float them in the new water for an hour along with adding a couple tablespoons of the new water into their cup every 10 minutes. After the hour I pour them in letting as little of the new water in as possible. You can train your betta to swim up out of the cup without using much water, over time.
While you are cycling you need to be testing daily for ammonia and nitrites. You should do a change any time either hit as little as .25ppm (meaning no purple and no green). In addition to this a weekly water change of 50% with gravel siphon is needed. The kit should be a drops kit like this one: http://www.amazon.com/API-Freshwater.../dp/B000255NCI
because strips are inaccurate. First you will see an ammonia spike, then a nitrite spike. Ammonia will stick at 0 first then nitrites. After a month or two you will be left with only nitrates that can be removed with a weekly 50% water change but gravel siphon. You need to siphon the gravel and not just change water off the top.
What filter are you using and how are you maintaining it?
Can you share photos or a video?
Food should have the first two or three ingredients as whole
fish, not fishmeal or the worst is wheat. I like New Life Spectrum Betta. I feed 5-8 (8 is high 5 is pretty minimum as these are small) spit up into two or three small meals a day with a fast day a week). Omega One betta Buffet pellets and Atison's Pro are other good ones. You could feed 3-4 split up with a fast day a week.