Well, the most direct route is below, not sure how much time you want to spend on this part of your trip, but going south to Grand Teton National Park and over has some nice riding.
Most Direct Route
Time Mile Instruction For Toward
Summary: 452.4 miles (8 hours, 11 minutes)
9:00 AM 0.0 Depart Yellowstone National Park on Local road(s) (North-East) 2.4 mi
9:05 AM 2.4 Turn RIGHT (South-East) onto Grand Loop Rd 8.1 mi
9:19 AM 10.6 Turn LEFT (East) onto US-14 [East Entrance Rd] 211.5 mi
1:33 PM 222.0 Turn RIGHT (East) onto Ramp 0.1 mi
1:33 PM 222.2 Merge onto I-90 [US-87] (East) 228.8 mi
4:39 PM 421.2 Entering South Dakota
5:07 PM 451.0 At I-90 Exit 30, turn off onto Ramp 0.2 mi
5:08 PM 451.2 Merge onto SR-34 [SR-79] (East) 0.9 mi
5:10 PM 452.1 Turn RIGHT (South) onto 4th St 0.2 mi
5:11 PM 452.4 Turn LEFT (East) onto Willard St 43 yds
5:11 PM 452.4 Turn LEFT (North) onto Local road(s) 54 yds
5:11 PM 452.4 Arrive Sturgis
SUMMARY
Driving distance: 452.4 miles
Trip duration: 8 hours, 11 minutes
Driving time: 8 hours, 11 minutes
Thru Grand Teton
Time Mile Instruction For Toward
Summary: 591.0 miles (1 day, 3 hours, 22 minutes)
DAY 1
9:00 AM 0.0 Depart Yellowstone National Park on Local road(s) (North-East) 2.4 mi
9:05 AM 2.4 Turn RIGHT (South-East) onto Grand Loop Rd 8.1 mi
9:19 AM 10.6 Bear RIGHT (South-West) onto US-14 [Grand Loop Rd] 20.6 mi
9:44 AM 31.2 Turn LEFT (South) onto US-89 [US-191] 44.6 mi
10:38 AM 75.7 Turn RIGHT (South) onto Teton Park Rd 8.6 mi
10:53 AM 84.3 Bear RIGHT (North-West) onto Jenny Lake Rd 0.4 mi
10:54 AM 84.7 Turn LEFT (South) onto Local road(s) 142 yds
10:54 AM 84.8 At Grand Teton National Park, return North on Local road(s) 142 yds
10:55 AM 84.9 Turn RIGHT (East) onto Jenny Lake Rd 0.4 mi
10:56 AM 85.2 Bear LEFT (East) onto Teton Park Rd 8.6 mi
11:11 AM 93.8 Turn RIGHT (East) onto US-89 [US-287] 4.0 mi
11:16 AM 97.8 Turn LEFT (East) onto US-26 [US-287] 155.6 mi
2:23 PM 253.4 Turn LEFT (North) onto US-20 [Idaho St] 65.6 mi
3:42 PM 319.0 Continue (East) on US-16 [Big Horn Ave] 64.4 mi
5:00 PM 383.4 End of day
DAY 2
9:00 AM 383.4 Stay on US-16 27.7 mi
9:33 AM 411.1 Continue (East) on Ramp 0.1 mi
9:33 AM 411.2 Merge onto I-90 (East) 178.4 mi
11:50 AM 559.8 Entering South Dakota
12:19 PM 589.6 At I-90 Exit 30, turn off onto Ramp 0.2 mi
12:19 PM 589.8 Merge onto SR-34 [SR-79] (East) 0.9 mi
12:21 PM 590.8 Turn RIGHT (South) onto 4th St 0.2 mi
12:22 PM 591.0 Turn LEFT (East) onto Willard St 43 yds
12:22 PM 591.0 Turn LEFT (North) onto Local road(s) 54 yds
12:22 PM 591.0 Arrive Sturgis
SUMMARY
Driving distance: 591.0 miles
Trip duration: 1 day, 3 hours, 22 minutes
Driving time: 11 hours, 22 minutes
North Route Thru MontanaTime Mile Instruction For Toward
Summary: 562.3 miles (1 day, 1 hour, 39 minutes)
DAY 1
9:00 AM 0.0 Depart Yellowstone National Park on Local road(s) (North-East) 2.4 mi
9:05 AM 2.4 Turn LEFT (North) onto Grand Loop Rd 7.2 mi
9:18 AM 9.6 Turn LEFT (West) onto Norris Canyon Rd 11.5 mi
9:38 AM 21.1 Turn RIGHT (North) onto US-89 [Grand Loop Rd] 78.8 mi (You can also take 212, a smaller road, never been on it)
10:05 AM 44.1 Entering Montana
11:14 AM 99.9 Turn RIGHT (East) onto Ramp 0.1 mi
11:14 AM 100.1 Merge onto I-90 [US-191] (East) 112.5 mi
12:58 PM 212.6 At I-90 Exit 446, turn off onto Ramp 0.5 mi
12:59 PM 213.1 Bear RIGHT (North-East) onto Laurel Rd 1.9 mi
1:02 PM 215.0 Turn LEFT (North) onto Access St 0.1 mi
1:03 PM 215.1 Bear LEFT (North) onto 6th St W 0.9 mi
1:06 PM 216.0 Turn RIGHT (East) onto Alderson Ave 109 yds
1:06 PM 216.1 At Billings, stay on Alderson Ave (East) 0.7 mi
1:09 PM 216.8 Turn RIGHT (South) onto Division St 0.1 mi
1:10 PM 216.9 Turn LEFT (North-East) onto 4th Ave N 2.0 mi
1:15 PM 218.9 Turn RIGHT (South) onto US-87 [Exposition Dr] 1.5 mi
1:18 PM 220.4 Bear LEFT (East) onto Ramp 0.1 mi
1:18 PM 220.5 Merge onto I-90 [US-212] (East) 3.5 mi
1:21 PM 224.0 Turn off onto Ramp 0.3 mi
1:21 PM 224.2 Merge onto I-90 [US-212] (South-East) 234.5 mi
2:52 PM 322.3 Entering Wyoming
5:00 PM 458.7 End of day
DAY 2
9:00 AM 458.7 Stay on I-90 [US-14] 102.2 mi
10:06 AM 531.1 Entering South Dakota
10:35 AM 560.9 At I-90 Exit 30, turn off onto Ramp 0.2 mi
10:36 AM 561.1 Merge onto SR-34 [SR-79] (East) 0.9 mi
10:38 AM 562.1 Turn RIGHT (South) onto 4th St 0.2 mi
10:39 AM 562.3 Turn LEFT (East) onto Willard St 43 yds
10:39 AM 562.3 Turn LEFT (North) onto Local road(s) 54 yds
10:39 AM 562.3 Arrive Sturgis
SUMMARY
Driving distance: 562.3 miles
Trip duration: 1 day, 1 hour, 39 minutes
Driving time: 9 hours, 39 minutes