The most direct route is to go North on US 89 to Livingston and take I90 west to Seattle. A good spot to overnight is Coeur d'Alene, ID which is a 7.5 hour drive from Gardiner/Mammoth. Googlemaps probably says it's less but this is the route I drive to the park (I get on I90 at CdA) so trust me on the timing.
I90 is actually pretty scenic especially from Missoula over the mountains and Lookout Pass to Coeur d'Alene.
An option that's a longer drive but beautiful is to go ahead and pick up I90 in Livingston, head west. In Missoula pick up US12 west. this cuts thru town then goes south and turns west over the next major mountain pass south from Lookout Pass which is Lolo Pass. ** Follow US12 over the mountains (2 lane road following the Wild and Scenic designated rivers of the Lochsa and Selway) down the mountains to Kooskia where you'd be beside the Clearwater River. You could stay overnight in either Orofino (really only 1 hotel here) or Lewiston which is at the confluence of the Clearwater and Snake Rivers. Lewiston is bigger with a choice of hotels, however I'd recommend the Hampton (newest in town), the Holiday Inn Express (next to the newest) or the Red Lion. Mammoth/Gardiner to Lewiston is about a 8.5-9 hour drive time but the roads are good so easily done in a day.
The route the locals take from Lewiston to Seattle is to go north on US95 then take the split to US195 just past the top of Lewiston Hill (this is a long 9-mile hill with great valley views as you climb from the 740' elevation at the river confluence to the farming area called the Idaho Palouse at about 2500'). Take US 195 north thru Pullman WA to Colfax WA. Take State Road 26 west to Vantage WA where you'd get back on I90 to continue the drive to Seattle. This is about a 6 hour drive depending on where you are going in Seattle and the traffic you encounter once you get over Snoqualmie Pass and into the Seattle Metro area.
Another option is to take I90 from Livingston west to Whitehall MT. Take Montana 55 south to MT 41 and follow this to Dillon MT. Get on I15 south for a few miles and take the exit for Wisdom MT/Montana 278. Follow this to Wisdom, then pick up MT43 west to US93. This goes past the Nez Perce BigHole National Battlefield from the Nez Perce War of 1877. Take US93 north thru the Bitterroot Valley to Lolo, MT and pick up US 12 to head west. You'd Follow the instructions above from the double **. This route is slightly longer and slower so I'd say it would be a good 9.5-10 hours from Mammoth to Lewiston this way.
That Best Western in Orofino, ID is about the nicest hotel in town. Actually, the only hotel as Pam said, but it really is a special place. I would go out of my way to stay there again.