In Mages Guild I was asked to steal a book from another Mage Guild member. The theft went horribly wrong and I got kicked out of the Mages Guild.

1. How do I get back in to the Mages Guild?

2. Does anybody have suggestions on how to do this theft better?

Thanks in advance for all replies.

Are you in Vivec?. You should talk to Trebonius and tell him you want to correct your mistake. After a few days or so, you will be forgiven. Or so I think.
And for the robbery ... Man, what I did was close the door behind me and loot the chest, and leave without talking to the owner. I think in the same mission, they even gave you some scrolls to open locks and everything.

Reload a previous Save file.
You do have at least one back-up Save file on the go, right?

Save all the time, on more than one Save file. Especially right before committing a crime on a quest.