Way, way back in 1987 - I finished school in May and worked all summer to save up for my first cross-country bike trip. Needless to say, Mom wasn't thrilled at the prospect. In fact, she said "I forbid it!" I responded that I wasn't asking her permission. Dad intervened to prevent WW III. So, I hear you.
Technology is such that you can be far more in contact than 25 years ago. But you should still consider having a fairly fixed itinerary rather than just wandering here and there. At least for your Mom's sake.
Not sure when during that 5 months you intend to do your riding. I'm also guessing that you need to do it on the cheap. But you should allow yourself sufficient funds so that you are not a total hobo - - again, for your Mom's sake. So will you be needing to work for a while to save up moolah?
Also, 1 month is way different than 2 months. If you are thinking closer to two months, why not ride across the entire U.S.? I mean, it's only about a week's riding from DC to Indy at the pace you suggest. Although, I would temper it the first few days so as not to burn out early. The reason they call it the Tidal Basin is because salt water comes up the Potomac that far.
Depending on the time of year, your choice in the West may vary. I would strongly argue against anything in the Southwest as temperatures are brutal in mid-summer. Even further north, there is a lot of desert terrain between the Rockies and the Sierras & Cascades.
If you want to meet up and ride with others, then I suggest using one of the Adventure Cycling routes. If you want more solo time, then you can pick just about any route you want - - provided that you do a little planning. Boulder is pretty urban - tough to ride into and out of, but not impossible. If you are planning on visiting friends - consider cycling to a nearby location and arranging a pick-up. Fort Collins is one possibility. The Poudre River Canyon west of Fort Collins on Hwy 14 is a sweet ride. Then again, you may want to ride Trail Creek Road in Rocky Mountain National Park - but be aware that you will be dealing with a good deal of traffic.
Winging it is fine in principle - but if you find yourself on a busy road with zero shoulders as night is falling and you are 20 miles from anything - then winging loses its appeal. Again - and your Mom will appreciate it - having a plan will let you choose lower traffic roads and actually enjoy the ride more. Plus, you can always vary your plans. But, I would take with a grain of salt any route suggestions you get from the bubbas at the local bar.
Most states have bike route guides on line which vary from excellent to so-so to poor. Most states also have traffic volume maps which indicate AADT (Average Annual Daily Traffic). I'll give you links to Nebraska's - - I prefer riding across Nebraska rather than Kansas. A bit cooler and more varied topography.http://www.transportation.nebraska.gov/docs/bicycle-guide-current-2.pdf
The color coding makes it really easy to spot the lower traffic roads - plus they show shoulder widths. Rarely, will light traffic roads have shoulders, but you don't need them. High traffic roads with wide shoulders are doable, just not very serene. I use a more stringent breakdown for traffic numbers than this map.
Under 500 - Super
500-1000 - Nice
1000-2000 - O.K. but more caution needed
2000-4000 - Tricky, shoulders really helpful
Over 4000 - Busy and shoulder almost essential
You can get the exact traffic count info here:http://www.transportation.nebraska.gov/maps/Statewide%20Traffic%20Flow%20Maps/2012-Statewide-Traffic-Flow-Map.pdf
As you can see, Hwy 92 in west-central Nebraska has almost no traffic and unplowed, rolling prairies.
Bests the heck out of hundreds of miles of pancake, dry fields in west Kansas. IMO
As for overnights, camping gives you way more flexibility. Often times lodging will not be available in places within reasonable mileage intervals - for ex. you may have the choice of doing 50 miles or 150. And camping will let you enjoy the West more - - not to mention that you can camp for free on most federal lands in the West provided you are 1/4 mile from developed sites.
Hope this helps.