As said, you will end up driving to trail heads. Unless you backpack overnight, there's no getting around that. You will also see some amazing sights on the drive. What's more important is that some of the best stuff to see in Yellowstone is in the back country. Lots of waterfalls and geysers that can only be seen in the back country.
Since your trip is far in the future, keep doing your research. The NPS has a ton of great info online. Additionally I would search for trail books that give detailed information about the trails, elevations, length, difficulty of hike, features, etc. Get more than one book and compare them. Carry them with you on the hike along with dedicated trail maps. Read these thoroughly then chose the ones you want to do most. You'll have a rough plan when you get there which will save you wasted driving and hiking time. See the quote in my sig? You want to enjoy the trip, not have an adventure.
It sounds like this is going to be a new experience for you. Get good hiking shoes, NOT sneakers. Been there, done that. Sneakers are bad in the back country. Get something along the lines of Merrells or Solomans. I'm on my second pair of Merrells, they are nothing short of excellent. Also hiking socks and hiking pants. None of this stuff is cheap but your feet and body will thank you for it later.
Regarding wildlife, the park has guidelines on that too. Something like no more than 25 ft from bison. It's unlikely you will encounter bears on the trails, but it's possible. If you are with a companion, keep talking, make noise. That will warn bears, they are actually frightened of people and will leave the area before you see them. You can get bear bells, but "word on the trail" is that they are not all that effective. And, yes, carry bear spray.
I wish you well, that's a great place to hike!
4runnerguy wrote:
I've seen several posts talking about a permit needed for backcountry hiking, but unless I'm missing something, I think a permit is only required if you are camping overnight in the back country. Am I wrong?
I was just looking to comment on this too. You are not wrong. We've day hiked the back country in several NPs. For day hikes, no permit is needed. Only for back country camping. We've run into rangers on the trails and have never been stopped or questioned.