Long time Colorado resident. Have been many times on all these roads.
Take I-70 through Denver and on to Rifle, turn north on HIghway 13, go up through Meeker to Craig, then take HIghway 40 straight to Vernal. Only a couple of small mtn passes between RIfle and Craig, nothing like on I-70, but be sure the roads are clear, as they can get icy. This time of year you should have no trouble. From Craig to Vernal is pretty much a straight shot with no passes, good road. DO NOT take the Berthoud Pass route - you'll be huffing and puffing over windy mtn roads and Berthoud Pass can be a bear. I-70 has two long climbing stretches (Vail Pass and the Eisenhauer Tunnel), but the lanes are wide and lots of places to pull over. Avoid weekend travel. Gas up in Rifle, not Glenwood Springs (expensive). You'll be fine, lots of big rigs take these routes. Lots of places to gas up in Craig, plus a big grocery store (City Market).
No need to go over Douglas Pass, it's a bit white knuckle and too much elevation gain/loss and windy roads. Keep it simple and go through Rifle to Craig to Vernal.
Will add that I live in Glenwood Springs. Nice hot springs here if you want to take a break. Right by the freeway, but parking will be hard for a big rig.