I'm trying to really max out my mage character, and I was wondering if I could join the Mage's Guild, House Telvanni, and Aundae vampire clan at the same time. Is it possible to be in all of them or do some hate each other too much?

Yes, it's entirely possible to be in all three and see each of them to completion. There are a few mutually exclusive quests for being in both the Mages Guild and Telvanni, but they're optional anyway.

Being a vampire can breed a few complications. Fortunately House Telvanni and the Mages Guild are still willing to deal with you, even if nobody else is.

You won't get any quests from the Mages Guild and very few from Telvanni if you are a vampire thus making it impossible to advance in rank. Plus any you get will be difficult to complete as any NPCs you have to interact with are more bothered about you being a vampire than with helping you.

Do Telvanni and the Mages together but forget being a vampire for the time being.

Thank you!

You can do Mage guild and house together with no problems and reach the highest rank in both around the same time, unmodded though becoming a vampire will make it impossible to get quests from either faction, so if you want to do all 3 save vampires for last and do the other two simultaneously. Also: the rewards from the other two factions will make life as a vampire more manageable anyway.