A quick check...it's 2185 miles via 75N o 24 to I-70/I29 north to I-80 at Lincoln, Nebraska to 287/26 in Wyoming into Grand Teton NP at Moran Jct. then up through the park into Yellowstone - if the south entrance of Yellowstone is open at that time.
Another route is 2344 miles via I-80 to Green River, WY then pick up US30 at Little America/Granger, WY over to I-15 at Pocatello. Then north on 20 to West Yellowstone.
Your route will depend on park entrance openings at that time of year and more importantly, on the weather. April can bring some snowstorms in the western states.
The safest route - weatherwise - would be 3070 miles via I-16 to I-20 to I-10 thru Tucson/Phoenix, AZ (spring flowers in the desert, outdoor Desert Museum in Tucson, nice hikes). Then north on I-17 to Flagstaff. Here is where you could do some nice touring of Grand Canyon then 89N through Page, AZ and do Zion and Bryce National Parks. Then at Panguitch, UT near Bryce, take SR20 over to I-15 and north toward Salt Lake City. (SR20 is a very good way to get to I-15 with a big rig. Don't take Hwy 14 off 89 - steep and curvy)
Good luck and enjoy your summer job! How many folks can say that they'll be living in Yellowstone? Neat.