The chainrings are on the front and the cassette is on the back. Just in case youre getting a little confused- here's a quick rundown...

What you have now is a "standard" chainring/crankset. It's got a 39 tooth small ring and a 53 tooth big ring. The diameter of the circle of bolts (AKA bolt circle diameter or BCD) that hold the rings together with the crank arm is 130mm.

A "compact" is a set of rings that is smaller. Usually the small ring will be 34 or 36 teeth (sometimes less, but 34 or 36 are most common). The big ring is almost always 50 teeth. The reason why you can't just change the rings from standard size to compact is the BCD- on a compact set of cranks, it's 110mm instead of 130mm. This is why it will be expensive for you to change to a compact- you have to get a whole new set of cranks with a 110mm BCD.

The cassette, on the other hand, is that group of gears in the back. Though they look complicated, they are relatively easy to remove and replace. Usually, you replace the whole cassette at once rather than just replacing the individual gears. It's a much cheaper endeavor than replacing your crankset.