Between about 9:30 am and 3:30 pm, the park is packed. Earlier is great, the people staying outside the park haven't made it in yet, and in the afternoon, they're all trying to leave. Not a great time to head towards the exits. Traffic can be like a big city. Also, many of the parking lots for the popular attractions fill up between 10-3. It makes it tougher to see the popular sights. Again, earlier or later and it's easy.
If you're into animal watching, get up very early and head to Slough Creek Road off of NE Entrance Road. Get there early, it will be crowded. In a 30 minute span, I saw a bison, bald eagle, grizzly, and a pack of wolves. There is/was a wolf den near the intersection of Slough Creek Rd and NE Entrance Rd. There's another wolf den closer to the Slough Creek Campground.
I am an early riser, so I was out early each day. I watched Old Faithful at sunrise. There wasn't another person in sight. Show up after 10 am, it will be thousands of people watching.