Welcome to the wonderful world of Betta keeping.......
In your current tank-half gallon-I would make every other day 100% water changes
Before you place the Betta in his new tank-if he came in a cup-save this and use for water changes and moving him around.
To move him to the new tank-put him and enough water from the half gallon in his small cup to cover him and then add some of the water from the new tank every 5 min or so until it is full-net him out and place in the new tank.
In a 2.5g filter tank- water changes of twice weekly 50% should keep water quality at safe levels, provided that you don't over feed and uneaten food is removed after feeding.
With the filter running 24/7 (required) the tank will cycle within 4-8 weeks, however, due to the needed water changes in that volume of water-cycling is a non-issue of sorts, but the tank will cycle and this will give you some wiggle room so to speak when life happens giving you an extra day and the fish still safe.
The filter media needs a swish/rinse in old tank water 1-2 times a month and when the water flow slows to get the big pieces of gunk off to maintain good water flow
You also need to clean the gravel with one of the twice weekly water changes
You can use either a turkey baster, siphon or a plastic cup-
First-unplug the filter and heater and give the gravel a little stir to get the debris up in the water column and remove half the water by dipping with a cup or siphon
Then re-fill with like temp dechlorinated water and plug the heater and filter back in-the water should clear in an hour or two
This need to be done one time a week and the other twice weekly water change can be water only.
Always wash you hand before and after you do anything with your fish and tank
It is important to keep the water temp within a couple of degrees from new and old water to prevent temp related problems and use a good dechlorinator with any new water added to the fish if on city water supply
If the Betta every acts odd or has a behavior change-make a 50% water only change even if you just did a water change the day before or earlier in the day.
In 2.5g tanks it is often easier to use a cup and dip the water
A good varied diet is needed-as posted earlier freeze dried bloodworms should only be fed 1-2 times a week as a treat, a good staple pellet made for Bettas should be fine
Remember their stomach is about the size of their eye and overfeeding can cause problems-small feedings twice a day is usually fine
Things you need:
bucket-labeled "fish only"
plastic cup or siphon
turkey baster (to use to remove uneaten food between water changes)
Dechlorinator if on city water supply
You can keep all cleaning supply in the bucket
You also need:
Heater to keep the water temp stable at 76-80F
Live plants will help and the Betta will love you for them....
I am sure I forgot something....but I am sure someone else will add as needed...
Congrats with your new wet-pet and look forward to hearing more about your Betta and seeing pic........