There's 3 types of guns/markers: pump, mech & electro. Pump means a pump handle is used to recock the gun before each shot. Mechanical (mech) is semi-auto, meaning it recocks itself after each shot. Electronic (electro) is also semi-auto but the trigger function is controlled by a circuit board and typically has other firing modes as well, like burst shot, full auto, and in other cases there's different tournament modes, to quickly set a gun to specific tournament specs. Electronic guns still need a gas source like all other paintball guns, which is explained below.
Compressed air and co2 are gas sources, it's what creates the gas pressure inside paintball guns, allowing them to operate & allowing them to propel a paintball. They both do the same thing, just are comprised of different gases. Co2 is carbon dioxide, obviously, while compressed air is simply the air you breathe, highly compressed.
Co2 fills are mostly liquid co2, which constantly converts to a gas that the gun can use. The downside to this is the liquid portion can sometimes get into the gun, and the gun does not like that. It's extremely cold, freezing & destroying o-rings (rubber seals inside the gun that prevent gas leaks), and in higher end markers, can destroy more sensitive internal parts. Co2 isn't actually dirty, just the damage it's capable of gives it the nickname "dirty," like "a dirty rotten scoundrel of a gas." Co2 tanks come in various sizes that are identified by weight, like 9 oz, 12 oz & 20 oz.
Compressed air, on the other hand, is just plain ol' air, so it's incapable of this type of damage, making it the prefered gas source. The downside is compressed air tanks cost more and not everyone has access to refills, since the air compressors needed to fill them are really expensive. However, they are ultimately cheaper than co2 in the long run because compressed air refills have no re-occuring cost. Compressed air has different names, like hpa, no2, nitro, etc, but they're all the same. These tanks also come in various sizes but also in 2 different materials - metal alloy tanks and carbon fiber tanks. Carbon fiber is lighter and comes in a greater variety of sizes, but they're also much more expensive.
Some pump guns & all paintball pistols use 12 gram co2 cartridges. Really no ups or downs to it, that's just what they use. Pump guns don't shoot fast so players typically carry less paint, thus needing less gas, so some just use the cartridges. Pistols simply hold less paintballs, so they don't need a larger gas source. One of these would be perfect for you, however pump guns and pistols are relatively expensive.
For your situation, you don't need to spend a lot of money. A cheap gun & a small co2 tank are all i'd recommend for you. May even want to check craigslist or ebay for used gear.