You've come to the right place. There's a lot that you can do on your own. Even if all you end up doing is gathering more information for the mechanic who ends up working on the car, they never mind a hint as to where the problems might be!
What are some of the possible causes for the "check engine" light that
can be diagnosed at home?
All of them. They can be all over the map, ranging from things as simple as "the gas cap is loose" to "major engine parts have broken off."
Where should I start?
You should start with a tool that will help you determine what the singularly unhelpful check engine light is actually trying to tell you: a code reader.
A code reader will query the engine computer to see what's on its mind. The engine computer will respond with a series of codes (from one to an arbitrary number of them), each of which will stand for a short diagnostic message. Depending on the tool that you use, you might also get a paragraph description and background information to go along with the code. Some tools will even let you clear out a code to see if it comes back (much like rebooting your PC to see if the problem goes away).
If the tool doesn't give you much more than some letters and numbers, try typing the results into the search box in the upper right of this page. There is a 100% chance that you are not the first person to have a particular problem and some of those people have come here with the same question. For example, I've had a P0420 code on my wife's old Outback. That lead me to conclude that one of the O2 sensors in the exhaust system was going bad. A little cash and a fair amount of annoying wrench work and the problem was solved.
If searching isn't helpful enough, come back with a new question letting us know the make and model of your car, the codes that you've found and any other symptoms that you've observed. This is a place where car nerds will happily help you for the fun of it!
PS: I just ran across Lynn's answer to the question What CAN'T the check engine light tell you? Lynn gives provides some links to the Wikipedia list of OBD codes as well as some examples of where those codes might lead you in diagnosis.