I had trouble posting this. Hope it's doesn't wind up as a duplicate post.
I'm from Europe originally, and I mention that because of the differences in weather in the summer. I live in North Carolina, which is a great place to start if you were to be tempted to do more than the Trans America route, rather than the northern tier. The thing to anticipate is the hot, but especially muggy weather. It really helps to have a tent with good flow-through ventilation. The other thing I've experienced personally, as someone who likes to stealth camp or camp at sites (so I don't have to impose a schedule on myself every day) is that the further to the east, especially the North East the more the traffic and the more difficult it is to find good free spots. Camping prices have shot through the roof over the last 10 years or so. I've stoped touring through Virginia because they have a out of state fee and one can pay up to $40 per night to camp. That adds up quickly if you're on a budget.
I normally like to find my own way around, but would strongly advise the use of the Adventure cycling maps and paths. In 2012 my twelve year old daughter and I cycled along the trans America up to where it met the Great Rivers South and then we took the Souther Tier portion down to finish in Austin. I was given one map and liked the design but arrogantly though I can save the money and gain more flexibility as well. What a disaster. We discovered that American lives and dies by the major highways. Out away from services all we saw on those nice less congested roads was gas station that had been closed for decades. Getting water was terrible and we found that away from where people are used to seeing cyclists, that while still mostly nice, that they were very much more suspicious of us and reserved. Once we changed plans and met up with the Trans American portion we found cyclists, services for cyclists and people along the route actually stopped the car a few times and got out to give us cold drinks and invite us to their home. I've learned my lesson there. ACA maps are really well thought out and I really enjoy meeting the cyclists one is bound to bump into.
On the topic of showers and washing, we found that we adapted and slowly did less and less of each. We also got rid of some clothing and I ordered wool. On me it made a large difference. By the end of the day my non-wool clothes reeked, where as the wool clothing took days instead. We would wash our clothes not according to a set schedule but more where we found it convenient. Sinks and campgrounds worked well and I like to use a bar of coal tar soap. It seems to help a bit with the odors and served to wash us as well as the laudry. One bar lasted the 54 days. We only used one laundro-mat and one marvelous Cycling hostel where the host did every cyclists laundry included with the lodgings and food price. That was a highlight.
We navigated with a garmin on each bike, powered by our dynamos, which ran through a reserve battery. It's great to not need to plug in often, but we often wanted to have a nice late lunch and chose a restaurant on most days even though I brought our Trangia cooking kits and food good for several days. Next time if I travel in the same conditions I will not pack food, though I will still probably bring a small alcohol shove to boiling water. We had temperatures that did not dip below 31 Celcius for about three weeks ( that was the night time low) and most often it was in the 36 to 42C temps in North Carolina and Kentucky. We completely lost our appetite at those temps unless we were inside enjoying mankinds greatest inventions. Air conditioning and ice-cubes.
We both experienced frustrating breakdowns with both garmins and next time I will use my phone instead. We also brought an iPad for my daughter's blogging, which when we had service (spotty in the mountains) was great for google maps in satellite mode, to find stealth camping stops up ahead. Most other cyclists used Warm Showers frequently, but we found that it went against our slow rambling style of travel. We didn't like to have to rush.
The other strong surprise for me was how generous the churches in the South East are, and how numerous. We were never turned down, often fed, and met wonderful people, by using the generous spirit of the church members.
As has been mentioned, not every day is wonderful, but we did not experience a single day where we had quitting even cross our mind. There is such beauty that the suffering in the hills, the sudden afternoon lightning storms, flat tires and heat - simply made it an adventure which we now treasure more FOR the fact that it was tough. You will find your own rhythm over the first two weeks or so. That first bit can be the toughest. Be kind to yourself with rest days or half days - and the adventure, the beauty, the freedom - it will keep you going. As will the constant comment from people you meet saying - wow, I which I can do what you are doing one day.
A last detail to mention is - buy good strong wheels and bring a few extra spokes. The only person we met along the way who had to quit was one who's wheels broke several spokes on the freewheel side, turned the wheel too wobbly to ride and there was nobody with a chain whip to get his bike back in shape for several days ride.
I think planning is half the fun. The interactive route map here sure serves to get my cycling appetite up. Good luck.