Sounds exciting.
This kind of fishing trip is not something I have done, but I have made many cross-country drives, and I am familiar with the geography of what is probably the most straightforward route, following I-90 or I-94 through ID, MT, the Dakotas, then MN/WI/IL/IN/OH, and then I-80 or I-76 through PA, and into NJ.
Following 90 out of WA, you go right past Lake Coeur d'Alene in ID, which has all three of your target fish.
Through ID/MT you'll pass a lot of mountain trout water. Your next opportunity for Trout will probably not be until PA, unless you choose to stay on I-90 at Billings, MT, and go into WY, and then the Black Hills, in SD. The other choice is to follow I-94 into ND.
You would cross the Missouri River in either North or South Dakota. The Missouri has a lot of pike, and good smallmouth in places. My wife's uncle fishes the Missouri river in SD for walleye a lot, and catches quite a few big smallmouth as bycatch.
Following I-94 through eastern ND takes you over some rivers with smallmouth and pike opportunities. Once you cross into MN at Fargo/Moorhead you then go through an area with lots of small natural lakes, around Fergus Falls and Alexandria, all of which have pike and bass. Continuing to the Minneapolis area, you pass just north of Lake Minnetonka on the west end of the city. Mille Lacs Lake is a little more out of your way to the north in the same area.
If you follow i-90 instead from SD, it takes you into the south end of MN which has not quite as many opportunities, but does have fewer people and less congestion. The freeway takes you just a little north of Lake Okoboji in Iowa which is one of the best lakes in IA for bass. Down the road, at the MN/WI border, you cross Pool 7 of the Mississippi River, another prolific bass and pike fishery
Generally, going through MN and WI takes you past tons of smaller lakes, virtually all of which have largemouth bass and pike, and some have smallmouth. I-90 and I-94 converge in Wisconsin, and you follow 90 into Chicagoland.
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Just after Chicago on I-90 you pass the southern tip of Lake Michigan, and there is some access at NW Indiana parks. Lake Erie is a little farther along, halfway through OH. At Toledo, you can take a detour north to Lake St. Clair. If you want to get on some Great Lakes smallmouth, these may be your best opportunities, unless you go quite a bit farther out of the way.
Once you get into PA, you re-enter some trout territory. Halfway to the NJ border, you'll come across the Susquehanna River, as @Susky River Rat mentions, likely at Harrisburg. Legendary Smallmouth river.
And then there's the Delaware River, on the border of PA and NJ.
These are just the most obvious places I know of, and I can't help with specific access points, but hopefully gives you as start on some places worth looking into.