October is a wonderful time to go. Typically we go a few times throughout the summer, however last year we did spend a few days up there in October. It's great because the crowds have died down quite a bit.
Plan to drive a lot. Depending on where you are staying, especially outside of the park, you will drive a lot. Both loops (north and south) have their own unique features. South loop is by far the most popular due to more features. I like to describe the park as a lazy man's park. Most of the features are accessible by car with a parking lot. We don't typically hike there. There are better parks for hiking. However, if you are wanting to stretch your legs, do the boardwalk at Old Faithful. Go all the way to Morning Glory Pool. Walk the observation view to see Grand Prismatic. Mud Volcanoes (Dragons Mouth, Black Dragon Caldron) is an easy loop. Artist Paintpots is a flat walk. Both sides of the canyon offer some great views with short walks. My favorite is Brink of the Upper Falls.
With you being there a week, definitely plan to spend time in the valleys (Lamar and Hayden). Those are the prime animal watching locations. Best times being early morning (sunrise) and late evening. You'll see people setting up spotting scopes to watch for wolves and grizzly. We've seen the big animals in both. Our last trip we saw wolves in Hayden and watched a grizzly scavenge for a few hours in Lamar. I recommend a spotting scope if you have one but at the least a good pair of binoculars. If you see a large crowd of vehicles (20+) stopped, it usually means a predator animal.
Traffic typically gets bad in Hayden valley because bison are constantly crossing the road. This time of year, it should be lighter.
Expect facilities to be closed. The only thing open in Canyon was the visitor center. Old Faithful will keep more open.
Take the side roads. Firehole Lake Dr, Firehole Canyon Rd, Blacktail Plateau Dr. Those are often skipped but have some good views and fun stuff to see without many people.
Drop down into Tetons as well. It's been our favorite park.