How much time and what do you want to see on which coast?
To East Coast from southern Arizona you could take I-10 (or US-90 or US-98 where available as more interesting alternatives with more to see) to Jacksonville. Guadalupe Mountains, San Antonio, Goliad, Houston, New Orleans, Mobile, Pensacola would be interesting stops enroute.
You can go up the East Coast on US-17 to Hampton Roads, US-13 to the Delaware River, before getting dumped onto Megalopolis toll roads from New Jersey through Massachusetts. On up the coast from Boston on US-1 to US-2 (as one west-bound alternative) or into Canada to connect with the Trans-Canada Highway.
Going north you can catch Savannah, Charleston, the Grand Strand, Wilmington. From Wilmington an alternative route will take you through the Outer Banks on the way to Hampton Roads. Beyond Hampton Roads be sure to take in Colonial Williamsburg and the Jamestown Settlement. US-17 follows I-95 to Savannah, then I-95 moves inland to run higher on the Atlantic Coastal Plain, but doesn't go through much of interest until Richmond and on in to D.C and Baltimore.
Going west you have to decide U.S. or Canada. To stay in the U.S. you'll have to go around the bottom of the Great Lakes, so whether you start with US-2, US-6 or I-90, by the time you get to Cleveland you'll have merged with I-80 which is the NYC to San Francisco Interstate route. A route across Canada can take you through Quebec City, Montreal, Ottawa to pick up Highway 17 across Ontario to Sault Sainte Marie, from which a trip across the north shore of Michigan's Upper Peninsula will re-connect you with US-2 across the northern U.S. into northern Idaho. Or US-8 out of the U.P. will take you to I-94 north of Minneapolis, crossing the middle of North Dakota into Montana, where it connects with I-90 to cross the Rockies into northern Idaho.
If your destination is southern Idaho, you might be better off with a more southern route, I-80 out of Chicago to Salt Lake City, where you can pick up I-15 to go north.
You should pace this to use the whole winter to go across the southern part, maybe spend the winter in Florida, then slowly follow spring weather northward, so that your journey west in the northern part of the continent takes place next summer.
If your plan includes traveling up or down the West Coast, I can't help you, I have no experience there. All the routes I've suggested, I've followed at one time or another during the past 55 years.
Tom Test
Itasca Spirit 29B