I've never been out to West Yellowstone, but it doesn't seem too far of a drive to get into the park from the town. But Yellowstone is a pretty good sized place. Depends mainly on how much you want to see & do as to how long it would take someone.
We did a quick trip in 2009 using Mammoth CG as our base. We mainly just drove the Loops of the loose Figure 8 found in the park. We only stayed 3 days. But it was really too quick to really see it well. It was our 3 trip & had plans on going back the next year (which we did). I'd say one needs a least 5 days to really see the place & take time to enjoy it. But 7 would be better.
The Speed limit at most is about 45 mph, but often less. Traffic can be slowed by number of people there & by animal sightings as well. A bear on or near the roadway can stop travel. A Bison herd can stop it as well.
Norris, Madison & Old Faithful Areas are fairly close to West Yellowstone. It is some 50 miles or so to Grant Village/West Thumb Area. 55 miles or so to Grand Canyon of the Yellowstone. Mammoth Springs is almost 50 miles from West Yellowstone.
Be sure to drive the Grand Loop Rd through the Hayden Valley (usually see Bison there) from Canyon Area to Fishing Bridge Area. The Lamar Valley has been a wolf watching mecca since the wolves were reintroduced in 1995. It has Bison as well & we have seen bear there as well. This is in the Tower Area east toward the NE Exit of the Park.
What I have on
Yellowstone.
You might consider driving out the South entrance & stay a couple of days to see & enjoy the Tetons as well.
Then leave for Grand Canyon from Jackson. We used US 89 to get from the Salt Lake City Area to the Tetons/Yellowstone in 2010. It was a nice route which went by Bear Lake & Logan in UT. But one could also go via Evanston, UT as well to bypass Salt Lake City.
Bryce Canyon & Zion are also off US 89 in southern Utah on the way to the Grand Canyon. All worth a stop.