Generally, colleges aren't pet friendly.
I'd say that living at home would be most convenient (for both price and comfort) especially if you live near the university.
Living at college is good fun, very social, and a nice chance to branch out. Depending on which college, you might have to cook for yourself. You'll definitely have to clean your own room (a very small space) and do your laundry. If you haven't got much experience with cooking and cleaning, living in college is a great stepping stone to get a feel for what it's like to have to do that on a daily basis.
Living in college will also provide a massive support system; most residential colleges will provide things like academic support and pastoral care, as well as loads of social activities. You'll likely have something like an in-house library so it's a good additional resource as well.
I don't know much about getting an apartment, but it seems like an extreme version of living in college in terms of responsibility. You'll have to take care of things like gas, electricity, and water bills as well as clean a bigger space. The rent will likely be the same, if not higher, than living at college as well. But, if you're up for it, the independence sounds really great.