Well, I've done this exact route once, and never touched Steubenville.
I was starting from north, but assuming you are Downtown, this is how I'd do it:
* North shore trail, to just short of the old jail
* Doerr, L Preble, R Eckert, L McClure
* L Antrim, R Fleming (moderate climb, but brief)
* L Termon, continue across to McKees Rocks Bridge
* Cross bridge, bending right onto Island Ave
* L Tunnel Way, continue through tunnel and down to traffic light
* R Broadway (in Stowe Twp), continue to end of street
* R Pine Hollow Road (long light climb)
* L Fairhaven (Wendy's, Pizza Hut). Be taking the lane at the couple of traffic lights approaching this corner, and expect horns. You're in suburbia here, but it isn't all that bad. Fairly low speed.
* Fairhaven is a tough climb, but only 1/4 mile.
* R Clever, and continue a good long ways.
* Just after Beaver Grade Road peels in from the left, peel R onto Old Beaver Grade Rd
* Continue to end, then L onto Montour Trail
* Ride trail for most of a mile, until you have to cross a road to stay on the trail. To the right is FedEx Drive, to the left is the climb into Robinson Town Center and Mall at Robinson.
* Go up that hill toward the Mall. At the "T" intersection, it doesn't much matter which way you go; they rejoin in about a mile. Meanwhile, you're climbing a hill with hundreds of cars passing you at full throttle. It sucks, but it's only a mile.
* When you get to Applebees/Eat&Park, turn left. Trek is in this plaza.