Also thinking about bringing my cousins dog, would we be able to get away with not getting a pack to carry his food and bowls?
That's just one more thing you'll need to carry in a pretty small pack. Since you're a newbie at this and likely have no ultralight (i.e., low volume) gear you may be hard pressed to fit all your own gear into it. I hope in your case that your pack is an external frame so you can strap some things on the outside. I also hope your cousin has a pack as well or you'll really be hosed. If not, borrow one.
I'd keep it simple and leave the dog with someone. Give that someone your route plan so they know when to expect you and can contact authorities if you are significantly overdue.
Do NOT carry 3 gal of water. 2 1-liter bottles at the most (per person) is all you should need. You can treat one while drinking from the other.
It was written from the Boy Scout perspective, but
this paper should be useful for other newbies as well. You need to learn how to layer your clothing properly and that comes mostly with experience.
Even if these are highly used trails, know how to navigate and what your bailout routes are in case something goes wrong. Also good to know some basic wilderness first aid.