The biggest hassle is having the kids confined for the trip each day. The second biggest hassle is that they will not appreciate the trip: e.g., the Sierra Nevadas will mean nothing to them which may be frustrating to you.
On the other hand, the 1-year-old is probably still in diapers and the 3-year-old should be toilet trained, so you won’t be breaking any good habits. (Never pass by a rest stop. (I exaggerate a little. A quick view of I-80 seems to indicate a rest stop every 20 to 40 miles through several western states. You might try stopping every one to two hours.) Take the girl to the rest room and make both of them run for a few minutes. The trip will take a bit longer, but it will be a more pleasant trip.)
You might consider a portable DVD player that can plug into a cigarette-lighter for the 3-year-old. (Batteries go really fast on long trips). (I would not usually recommend a TV for older kids if you were travelling scenic countryside–they’d like it but it would drive you nuts when they ignored the scenery–but for a small kid trapped in a car for five or so days, it might be worth it.)
Yes, it is already hot by June. There may be more cool days than July or August, but it may also be more humid.
For camping, you might want to invest in the Woodall’s Guides ($9.95 each for the Western U.S. and Eastern U.S.). Rand-McNally used to publish campground guides that were a lot easier to use (and a lot less biased toward advertising), but they got out of the business a few years ago. The Woodall’s guides are still OK. They provide directions, lists of services, and ratings for most campgrounds in the U.S., state by state. You can also use Google™ to dig up campgrounds, state by state, but that would involve a fair amount of time.
You should not need reservations for camps Sunday night through Thursday night and would rarely need reservations on the weekend except around major attractions (few of which are along I-80).
For in-and-out cheap camping, few sites beat National Forests, wildlife areas, and similar locations that have not been granted “park” status. You might have to forego a shower that night and they will have few amenities, but they are generally cheap.
(BTW, if you have never camped before, you really need to take a weekend for a test run to see how you and the kids do.)
There are several low-cost motel chains out there, although “low” is a relative term. Days Inn, Knights Inn, Motel 6, and Red Roof Inn tend to run $45 - $70 a night. LK Motels used to be fairly cheap, but I am not even sure they still exist. Mom and Pop Motels can be cheap and wonderful or terrifying, dirty, and expensive and it is difficult to know in advance. Someone who uses Orbitz and similar services might be able to tell you better whether there is a way to find good, inexpensive motels. Similar to campgrounds, weekday nights you should rarely need reservations; weekends near large cities (epecially if you intend to travel late into the evening), they might be a good idea.
Are you planning to return the same way? Or is this a one-way journey? If you are going to travel the length of the country twice, I would strongly urge you to use separate routes for each direction unless speed is of the essence.
Essentials:
Wet-Ones
Paper towels
Zip-Loc or Glad-Lock bags, quart size and gallon size, at least. (Sandwich and snack bags optional. The 2-1/2 gallon monsters are not a bad idea.)
A good cooler. (As the ice melts, do not pour it out. The cold water still keeps things nice and cold. If you have cooled food that mustn’t get wet, put it in a locking bag.) A seven lb. bag will cost $1.00-$1.20 and last a day and a half.
A gallon thermos with spigot. Fill with ice, (about 1/3 of a 7 lb. bag) then fill with water, juice, or soft drink of choice. That should generally last one day unless you get really thirsty.
Carrots, celery, (radishes if your kids will eat them). Slice them into bite sizes each night, them place them in bags in the cooler to be passed out when the kids need snacks. (Unlike crackers, the veggies are less likely to make the kids more thirsty–and with fewer crumbs. String cheese is a little salty, but it can keep them occupied for several minutes if they learn to peel strands rather than biting it off like carrots.)
Sippy cups, insulated to hold ice if you can find them
Camera
Recommended:
Check list (be sure to put the kids’ toys on it and check it before you leave any place you stop)
Sun shade (the kids sit lower in the car and several hours in the sun (on your shoulders but their faces) will not make them happy people)
A carry-bag for the 3-year-old’s three or four favorite toys or books, with a rule that if one thing comes out of the bag, one thing must go in. (Yeah, I never could enforce that, either, but it does not hurt to try.) Similar bag for the younger child under your control. (NO toys with small parts.)