In a 10gal filtered tank without testing products-cycling with one Betta-I would make at least twice weekly 50% water changes-one of the twice weekly to include gravel vacuuming/substrate cleaning and one water only- for at least 4-8 weeks and then one weekly 50% with vacuum to maintain water quality after cycle is complete.
With testing products-make once weekly 50% water change with vacuuming and 50% water only with any reading of ammonia/nitrite 0.25ppm or greater...once you have reading of nitrate 5-10ppm and 0ppm on ammonia/nitrite for several days without water changes you are most likely cycled
I would continue to monitor the water prams and make the needed water only changes per the reading until the tank matures between the regular weekly water changes.
Don't over feed and remove any uneaten food after feeding, keep the water temp within a couple of degrees from new and old water with water changes and use dechlorinator with any new water added to the tank if on city water supple
Give your filter media a swish/rinse in old tank water with a water change 1-2 times a month and when the water flow slow to get the big pieces of gunk off to maintain good water flow.
Look forward to seeing pics......