I agree with almost everything Thor said. Except, as you've already kinda pointed out, maybe the heater. You need a heater if the tank is too cold. I live in FL, I don't have a heater. My house is rarely ever below 74, usually well over that. So, it is relative to where you have the tank. If it is going to be in a room that usually 68F, get a heater. Etc etc.
The "Bettas are lazy" idea also usually comes from people who keep them in a cup, lol. the Betta I have is actually quite active and interesting. And amazingly compatible in this tank.
I know you are trying to save money here, but I would recommend a test kit to at least test for ammonia, nitrite, & nitrate. pH would be a plus. Most of those API tests will last for years if maintained properly. And it will enable you to track the nitrogen cycle as it goes along. Until it completes (by this I mean 0ppm levels of the two most toxic compounds, ammonia and nitrite), add Prime to the water every 48 hours. This will detoxify the water as Thor said. But it only lasts a short while. Once those levels are at zero, you can then stop this process, and start regular water changes, only using the prime when adding new water. But again, if you have no idea where this process is at, you are just guessing, and risking the health of your fish.
FWIW, I just cycled a new tank up, with a decent number of "moderately sensitive" fish, using this process, with 0 deaths. Almost two months now. Knowing the nitrogen levels and using Prime accordingly, is rather amazing. I actually stocked more in the tank than I wanted, assuming something would perish. Now I have a well established cycled tank, and am moving fish around, lol! Unless you are only going to add highly sensitive fish, the fishless cycling is almost pointless provided you can monitor the levels and have Prime on hand. But in order to do a proper fishless cycle, you need to monitor the levels sooo... there is no real advantage.
I should also add, this is dependent on a filter with adequate biological media, and can vary with the species of fish. Bettas are famous for being able to adapt to water conditions that would kill almost anything else, even goldfish, hence why they are sold as they are in tiny cups.
