Like others have mentioned, just hop in the lane with the semi trucks and you'll be fine. We do this drive once a year to head to Country Jam, and have done it with all variety of trucks. I would also agree that going downhill is the harder part :)
I would also plan out your gas stops before you get on the road. I use Google Maps with satellite and street view to plan out where we are going to go. Many exits have gas stations, but as they are in mountain towns, many of them are difficult to maneuver with a trailer. There is a Sav-O-Mat station in Silverthorne just north of the outlet mall that works pretty well for us, we always top off there even if we don't really need to since it's a ways before you get to the next easy to access station. There's way more options once you get past Eagle, but most of the stations in places like Vail are tight. There is a Conoco in Vail off the South Frontage road towards the west end of town that works OK, just depends who all else is there. There is also a Costco in Eagle with good prices, just a bit off the beaten path but easy to get to and lots of room to move around. This won't matter for you but if anybody else cares, they have diesel there too, one of the few Costco locations around that does.
Also, if you need it, there are free water/dump stations ran by CDOT in Edwards, Rifle, and Fruita, easy to access right off I-70. So if you need water at your destination, these are nice as you don't have to haul a trailer with a full water tank over the mountains. They also have truck parking so you can take a rest and stretch your legs. We do that at the Edwards location but use Rifle for filling up. I've never seen either one crowded, but Fruita can get busy.