Hi FD,
Siamese fighting fish (or Bettas) are a native of Thailand (obviously from the name) and as such are a tropical fish, so firstly they need a fairly stable temperature (despite what you may have been told) and an aquarium heater would be a good investment. The acidity of the water should be around pH 7 (ie neutral) for most Asian tropical fish.....but it varies from species to species depending on the environment they come from. It's been a while since I kept fish but I used a simple pH test kit with a little test tube to sample the water and you add a few drops of an indicator fluid which changed colour according to pH. You compare the colour of the test tube to a colour chart. This only takes seconds and is cheap and reliable. You can spend heaps more on electronic meters...but really pH is only a part of the equation.
The main thing is to avoid big changes in water quality/temp. Keep things stable and make changes gradually. Chucking a heap of Sodium Bicarbonate in would not be very good, even if you are trying to correct bad water quality. Bicarb is a good buffer which helps stabilise pH, but too much can also send the water pH too high....and your water may already have enough hardness that will already keep the pH stable.
Fighting fish do have the capability of taking a gulp of air to supplement the oxygen in the water as they are often found in small pools and puddles in Thailand with low oxygen. Because of this they are reasonably tolerant of bad water conditions. But if you want your fish to have a long life it's best to ensure good oxygen in the water. Bubbles don't actually add much oxygen to an aquarium. It mainly comes from the surface of the water interacting with air. Bubbles will help remove carbon dioxide from the water though, and help circulate the water.
By the way, your filter is "aerobic" (requiring oxygen), not anaerobic. The job of your aquarium filter is to establish and maintain a colony of aerobic bacteria which will convert deadly ammonia from fish waste and respiration into less harmful nitrates. This is the same process that sewage plants use to treat sewage. And this leads into the golden rule of keeping fish. You need to regularly change the water to avoid a build up of nitrates which will kill the fish eventually. 25% every week or two is a good rule of thumb. If you don't change water a fish will eventually choke in it's own waste...a filter will only help for so long.
Finally, remember that chlorine in city drinking water is deadly to fish. Always use a chlorine neutraliser. Sometimes pet stores don't always tell you all this info because they quite like selling fish to replace ones that die
As you can see, successful fish keeping is a BIG subject. Reading a lot is a good idea if you want to keep fish alive.