Hi Heather, good on ya for your tour! I rode in NZ last February-March, and did a cross country trip here in the USA this past summer. I wish I had known about the great facilities at the Holiday Parks and Backpackers there, I wouldn't have bothered with a stove... although i used the heck out of my tent.
In the US, there is never a kitchen in a campground or RV park - if you want to cook, you must bring your own stove and everything else. Motels are generally at least $30, and more often $40+. Backpackers (we call them hostels) are very rare, and usually are just in larger urban areas. Camping here is generally charged per site, not per person, so it's possible to share to save money. RV Parks (most like Holiday Parks) are $15-25 and have coin-op laundry (about $3-4 for wash and dry). They usally are pretty ugly and noisy. National park/forest service campgrounds are $5-10, and usually just have water, pit toilet and picnic table. These were my favorite - they are usually in a beatiful natural setting, often with a river or lake to jump in. Some local parks (city, county) allow you to camp for free, I checked at town hall or the police station to see if that was ok. Stealth camping is possible, but I didn't do much, so I have nothing to recommend.
If you are going to camp, you will need your own sleeping gear and rain protection. Campgrounds don't provide anything.
Not all small towns have motels. I didn't do the TA route, so I can't speak for what it would be like to do that using only motels, but the route I took wouldn't have worked without camping.
I found camping and cycling gear to be a whole lot cheaper here than there, so you could try going light and then buy stuff here if you need it. Try REI.com and REI-outlet.com for good stuff, good deals, and overnight shipping.
Have a great trip!