I'm going to start a new character in Morrowind, but I have no idea what class I should choose. I know I will be playing as a male dunmer, and I want to join the ALMSIVI temple, and Great House Redoran. It also must be a class that doesn't rely completely on magic (but some magic is just fine), and does not involve stealth. Does anyone have any suggestions? Thanks

depends, do you care about optimization? If not, the Crusader class you can start off as is basically a Paladin and so has both the religious connection for the temple, and the Warrior stuff for House redoran.

If you do, then maybe someone else can make a more optimized version of the crusader class.

I pretty much always create a custom class.
You can check here to see the favored Attributes & Skills for each faction/house. http://www.uesp.net/wiki/Morrowind:Factions

The http://morrowind.inventivegamer.com/tools/chargen/ is also very helpful in building a class.