Thanks for all these notes! In terms of the route AZBuck, our RV return is in Linden, NJ. For specific interests, as many have alluded to, I think mainly places for the kids to get the energy out. Thanks so much again for all these notes!
I-80 will still form the backbone of your driving out west, but to avoid the toll roads out east, and the congested cities you'd be paying to get around or through, a slightly different route could pay off pretty well. The route laid out below adds just under a hundred miles but misses the Tri-State Tollway, The Indiana Toll Road, The Ohio Turnpike, and most of the Pennsylvania Turnpike.
You'd leave I-80 at Lincoln NE and take I-29 to St. Joseph MO, then US-36/I-72 to Champagne IL. Note that US-36 is just slightly below Interstate specs for most of its length it's four-lane, divided, limited access roadway. I-72 joins I-74 at Champagne so just follow that to Indianapolis where you'd take the Beltway (I-465) around the northern side of the city to I-70 eastbound on the other side. I've found that when driving relatively slow and unmaneuverable vehicles that the best thing t do in multi-lane, heavy-traffic situations like this is to get in the next to the farthest right lane rather than the farthest right. Leave that on for people getting on or off. Anyway, continue on I-70 east, using Beltways around larger cities where available, to Mount Pleasant PA and join the Pennsylvania Turnpike (I-76 TOLL) east to Bedford PA and hop on I-99 north up t0 I-80 east. That will finally take you into the Newark area and I-280 to the Garden State Parkway (TOLL) to almost Linden.
Even with an extra day or two, your time is still going to be your most precious commodity, so don't be tempted to take major detours to see "must see" attractions. It really won't take much travel off the most efficient routes to have you end up spending all your 'extra' time driving to places you then won't have time to enjoy. There's plenty along either of the two basic routes (I-80 all the way, or the slightly longer route outlined above) to keep you and the kids occupied, entertained, and informed without making things more stressful than they're already going to be.
I would modify Buck's recommendation as follows, this keeps you on Interstates and avoids PA Turnpike tolls:
Stay on I-80 to Moline IL, then take I-74 to Indy. Take the southern I-465 bypass back to I-70.
At Columbus OH, take the northwest I-270 bypass to I-71. Take I-71 to I-76 through Akron to I-80. To get to Linden, take I-287 to NJ-24 to I-78 to the Garden State Parkway. Go south to Exit 136. Note the the GSP has relatively low arched bridges in that old section, use the 2nd lane from the right if your RV is tall.