First of all, don't worry about the nitrogen cycle for now. You have enough to think about. In fact, don't even install the filter for a while. Get used to fishkeeping and tank maintenance.
Your instinct about water temperature is correct. If his current tank and his new one are at the same temperature, you can gently scoop him out of his tank with a clear cup (maybe the one he came in ...no nets) and put him into the new tank. To warm the water, reset the heater every couple of hours until it reaches 78*/80*. You need a cheap, floating thermometer.
Using your siphon, change 50% of your water twice a week. Use 10-drops of Prime water conditioner with each change. Fill a one or two gallon jar with tank-temperature water and the Prime. Match the temperature by running the hot/cold taps until it feels the same. (You can use your thermometer to match temps until you get the feel.) Pour it from the refill jar slowly into the tank. Maybe your fish will play in it while you pour. That's fun for everybody.
Feed him New Life Spectrum Betta pellets or Omega One Betta Buffet. Nothing
else until you know what you're doing. Many experienced keepers, including me, feed them nothing else. Feed one at a time by hand, about an eye-sized bunch of pellets once-a-day, or half that twice a day. Hand-feeding keeps a Betta tame and friendly, and it's a lot of fun. Remove any food he doesn't eat with your siphon or a new
turkey baster used only for this purpose. Remove any feces you can get to easily. Don't get carried away with this, it's supposed to be fun.
This will give you a running start:
Welcome to the forum.