Unless you need to filter multiple tanks from the one pump, any small, single outlet air pump should be fine for a betta tank. I purchased a cheap low end pump and it has worked beautifully for over a year. It does struggle if I start to try and hook up more than a couple of tanks but that is to be expected as it was not intended for this purpose.
A check valve is just to stop your tank from back siphoning into the air pump if the power goes off. I don't use these (naughty) but any one will do. They probably all come from the same factory in China, so as long as it can fit inside your airline tubing it will be fine.
A control valve obviously allows you to control the flow of your sponge filter. I like to use valves that look like this:
I find they provide the most precise level of control. Some types of control valves only seem to allow you to have lots of flow, or no flow at all. There's no real in between.
I personally like the soft, silicone airline tubing. I find it is easier to attach and remove when compared to the harder plastic tubing. Colour is personal preference. I like using clear as I feel it is less obtrusive.
Once you have purchased all your equipment, you basically attach a length of airline tubing to your air pump, insert both your check valve and control valve somewhere along the tubing (you just cut where you want to place these and add them in) and then attach your tubing to your sponge filter.
Giving the sponge filter a few quick squeezes will get the air out and make it sink, and then you just turn your air pump on, and adjust your control valve until you are happy with the level of flow.