The fan comes on, but the freezer and refrigerator are not getting cold. Should I check the relay for the compressor? Or just change out the compressor?

Relay is a lot cheaper $20-25. Check for 120 volt continuity to the compressor.

Look at this: https://dl.dropboxusercontent.com/u/8010163/How%20to%20use%20Test%20Equipment%20Service%20Manu al.pdf
https://dl.dropboxusercontent.com/u/8010163/Compressor_Data_Manual.pdf
You use an ohm meter and disconnect all wires and the start relay from the compressor terminals. Then you check all terminals to a metal spot on compressor and change ohm scale to highest scale which should be mega-ohms. If you get a reading to ground then the compressor is bad. Make sure to read from a copper line where you can make good electrical contact. If it passes that test then reads on the lowest scale between each terminal. You have three terminals: Common, start, and run (or motor). The highest reading will be from start to run; second highest from common to start, and the lowest from common to run. Sometimes they are labeled, but you always use your meter to know which one is which. Many times the top terminal will be common, but either one of the bottom ones can be start or run. This applies to single-phase household refrigerators and larger ones as well. Three-phase units are all equal.

You can buy an electronic relay with start capacitor to replace a bad relay and fix a unit with a compressor that is hard to start or has become stuck. Unplug the relay and if it has a coil of wire on it then you have a current relay. By turning it upside down it will close the contacts between the two terminals. They are also designed so if you clip a jumper you can put in a capacitor. If it does not have a capacitor then it just closes the connection between the two terminals it plugs onto.
Check out the compressor, too. Look at these: https://dl.dropboxusercontent.com/u/8010163/Compressor_Data_Manual.pdf
https://dl.dropboxusercontent.com/u/8010163/How%20to%20use%20Test%20Equipment%20Service%20Manu al.pdf


Here are some Maytag manuals: https://dl.dropboxusercontent.com/u/8010163/Maytag%20Refrigerators.zip