If you are really low on budget, the cheapest filter system available is sponge filter. You can get them for around $10 from Amazon. I recommend Hydro Sponge filters. The downside is they take space inside the fish tank, and they are not that good looking lol. Nothing ugly, just not that good looking.
It is true that you can skip air pump if you use a power filter in many cases. Since the purpose for air pump is to create bubbles and when the bubbles are busted at the surface, they create surface movement which will promote gas exchange between the water and the atmosphere. The waterfall from the power filters do the same thing - create surface movement.
However, if you go for a sponge filter, you will need either an air pump or a water pump.
A 10 gallon fish tank is not enough for one goldfish, let alone two. Comet are only slightly smaller than common goldfish. You will have a hard time to keep the ammonia at 0ppm with a 10 gallon tank.
For cycling the fish tank, it is quicker if you use pure ammonia for fishless cycling (without fish). It is the fastest if you use live bacteria products to seed the filter media. I recommend Tetra SafeStart.
No bottom feeder will eat fish poop. However, bottom feeders will eat leftover fish food at the bottom of the fish tank. If you get goldfish, which are not tropical, your choice of bottom feeders are limited. If you go tropical, a lot more options are available. From corydoras catfish, to red cherry shrimps and snails. Fish poop has to be removed by water change. Filter does not do that job (even if it does suck in all the debris, it only removed them from your sight, but the rotting debris are still in the water system producing toxic ammonia).
Once the tank is fully cycled, and fully stocked (not overstocked), all you need is daily feeding (no more than what the fish can finish within a minute), and weekly partial water change of 30~50%. Filter maintenance is purely circumstantial. If you think there is too much debris sucked into the filter, or if the water flow has been slowed, then go for it, but be careful not to damage the bacteria colony in the filter media.
Only one betta fish per fish tank. Betta will fight each other to death. Also you must not to mix betta with other fish (except) bottom feeders.
