I've been reading around about classes and I noticed there is a lot of discussion on them, if I make mine wrong can it more or less make the game impossible?
Refer to the link in my signature, and it will explain everything you need to know about character creation. The short answer to your question is NO. You cant make a broken character in Oblivion, at worst you will make a counter-intuitive one, meaning you'll make one that winds up using skills that you didn't really want to use.
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've also been reading about an archer class, It sounds interesting but is there anyway to get some sort of animal companion to follow me around everywhere? It seems like it would make the game a lot more enjoyable if I had a pet that would help me out through it.
No you cant get a pet in vanilla oblivion, you can however cast a spell called "Command Creature" that will make animals fight for you for an amount of time.
I've also read about something called fast travel. What is this and should I use it or not? Is there a downside to using it?
Fast travel allows you to quickly move to towns and any pre-discovered locations. The only downside to using it is that you may miss alchemical ingrediants and have a harder time leveling up Athletics and Acrobatics skills.
Thanks for taking the time to answer my questions. Happy gaming!