An under-gravel filter provides a fair to midling range of filtration. Depending on the amount of water flow and depth of the gravel above the grid, changes how much true filtration you get with an UGF. They can also be a bit of an issue with live plants, depending on the gravel/sand mix (and other things as well) and water flow. The higher the flow rate, the better the filtration, but the more damage you may have with the plants.
A HOT (hang on tank) type of filter typically offers multiple filtration chambers allowing you to modify how it runs, meaning you can run different types of media inside, say lava rock for allowing growth of the beneficial bacteria, or that fibrous material that looks like cotton balls to allow for solid filtration, meaning it helps to remove the floating solids. With the ability to mix and match the media AND the ability to only change a part of the media each month, meaning you might replace the fibrous stuff but leave the lava rock alone, you gain the ability to keep the good, living bacteria growing on the lava rock, while still removing the nasty solid wastes when you change the fiber media out.
There are multiple variations on HOT filters, offering just about any style of filtration you want. In my case, for all 3 of my tanks, I always have at least 2 different filters running at all times. I always have an undergravel (two run on air, the 3rd tank uses a power head) and then one of two different types of HOT style filters or canister style. Canister filters are the same as the HOT style, just much larger, giving me the ability to have many more media types in the same filter housing, thus allowing for more fish in a tank, and/or multiple ways to treat fish if they run into issues etc.
As to having just plain air bubbles... that is a complete waste of power and time UNLESS you are using those bubbles as the lifting power in the uplift tubes of the under-gravel filter. The amount of oxygen dissolved into the water of the aquarium is much greater just on the surface of the tank, and the air bubbles truly don't make enough difference to even be measurable. The only advantage they offer is that they make the surface of the water at the top of the tank, move a bit, which does increase oxygen content but that is also gained by almost any type of filtration... if the surface of the water moves at all, then it almost doubles how much air actually gets absorbed into the water. Your under-gravel or hot filter will move the water enough to make the air pump completely unnecessary unless you are using it to 'power' the movement in the under-gravel filter.