[COLOR="Blue"]Sounds like it runs fine[/COLOR]
Sorry but "Sounds like" is not specific enough.
When running you should be able to hear the compressor, feel it vibrating and it should be warm.
It makes a big difference on how to troubleshoot if the compressor is on or off.

[COLOR="Blue"]but Freezer and Frig won't cool.[/COLOR]
Also this is not specific enough.
Often people say it is not cooling when in actuality it is but is not cooling enough.
Best way to check is to remove the cover in the freezer so you can inspect the evaporator coils.

What are all the common possibilities I need to look for in determining necessary repairs?
Cannot say what the possibilities are without more info.

Here are 2 good sites with basic fridge repair/troubleshooting info.
http://www.applianceaid.com/frig.html
Refrigerator Repair Guide: How To Fix a Refrigerator - ACME HOW TO.com

If the compressor is running and there is no or blotchy frost (usually where the freon enters the coils) on the evaporator coils then:
1. The compressor is toast
2. All the freon has leaked out
3. You are low on freon.

Often this is the end of the units life as you need a pro to repair it and it will be expensive. Usually better to put those dollars into a new unit.