Some fantastic suggestions here.
I personally love the Highway 1 run along the western coast. Even if you take the more direct route from Washington (state)-->California, the roads are less dreary than some others with constant changes in vegetation, color, and weather. Plus, California has Hollywood as well as famous amusement parks if your kids are old enough (Magic Mountain, Disneyland, Sea World, Knottsberry Farm--listed in terms of preference when my friends and I were younger).
Washington is one of my favorite states in the US. It's one of the few places in the world that has every geological formation from Mt Rainier, rainforests, waterfalls, lakes to sand dunes. Try the clam chowder in a bread bowl in Pike's Place--yummm!
Portland, Oregon is the only city I particularly recommend in the Oregon. It's a cozy city that's often overlooked, but equally lush as Washington. A good place to relax and have a coffee/meal before moving on. It also has a nice Japanese garden and one of the largest independent bookstores named Powell's if you're into that. Plus, everything's tax free, yay!
San Francisco would be my favorite spot in California. There's too much to do and always too little time. You can also use yelp online to look up local favorites--I once stood out at the break of dawn, under the rain (and umbrella), waiting for space to open up in a breakfast joint, and it was wellll worth it.
I'm biased towards the Colorado mountain ranges because I grew up seeing them every day and never grow tired of them. If your children are younger, there are places like "The House of Bounce" that my 3 year old nephew loved, or The Butterfly Pavilion, small, but educational and nice. But, as Dave & Merry Jo suggested, the mountain ranges and ski resorts are one of the most famous places to go. My favorite is Breckenridge (good for skiers and snowboarders), but you really want to check the weather before heading to these places as the roads can become blocked with snow/ice and traffic.
Route 66 is a frequent historical drive taken by locals and visitors.
I'd recommend the south to hear the southern drawl and try a big plate of tasty greasy food. The coast of Texas is also famous and overlooked for shrimp.
Washington DC for history and beautiful monuments.
I don't recommend driving in NY, so flying there (maybe from Miami?) might be a good idea. I don't even like taking taxis there, and would recommend booking a good accommodation ahead of time and trying to walk or take short taxi trips as the traffic (human and vehicle) is pretty crazy.
Places like Colorado, Illinois, NY and Washington will be cold or even freezing (Washington's roads occasionally get black ice). As long as you drive slowly and patiently with an ear out regarding weather you should be fine. Avoid blizzards. 😊
If you go here
and look under the heading "The 1925 Routes," you will see many of the major highways you can take through the States today.
As for gas--big cities, especially the famous ones--tend to be more expensive. The Midwest is on the lower end. But, overall, I've noticed that gas prices tend to be lower than other places in the world, but I don't know how it compares to OZ. You can look up specific prices for states on this site
As with any travels, budget shifts based on taste. My personal preference is saving on food and accommodation (I agree with Sojourner 1208 on finding a Motel 6 in most states for about $50/night) and then splurging every once in a while.