Here is my current back packing and moto camping tent:
https://www.rei.com/product/128692/rei-co-op-half-dome-2-plus-tent
I find a "two man" tent to be about ideal for one of me. I like the rainfly design that has the outer vestibule areas for my boots, helmet, and rain gear.
Here is my current back packing and moto camping sleeping bag. I like the semi-mummy style, finding a pure mummy style to be way too confining, and a big square one to be unnecessarily big. Pick a temperature range based on what they rate it at, plus whether you sleep "hot" or "cold". For example, I'm down to my skivies with one leg hanging out of the bed, sweating, while my wife has her flannel night shirt, wool socks, and sometimes her wool hat on and she is still "cold". Get one that's rated warmer than the temps you expect if you sleep "hot", and cooler if you sleep "cold".
https://www.rei.com/product/127386/rei-co-op-trail-pod-30-sleeping-bag
Equally important is a dry bag, like they use for river rafting and stuff. Nothing worse than crawling into a wet sleeping bag after setting up a wet tent. Something like this:
https://www.rei.com/product/114970/nrs-bills-bag-dry-bag-65-liters
As far as where to pitch your tent, I've had experiences similar to Bob's - too many dicey spots where I wound up wet or worse. I now stick to state parks or national parks. I've never reserved in advance and have never had any trouble finding a spot for a bike. Granted, I avoid holiday weekends, preferring not to compete with the motor homes and campers on the road or for a spot in the campground.
I have had good luck with KOA's as well. I like their Camp Cabins, usually a one room, two bed little log cabin affair. These have saved me in a downpour on many occasions.
So, if you grab a few parks guides to include the states you will be traveling through, a national parks, and a KOA, you should be set. There is always one or the other somewhere close to your intended route. This has worked for me for 40 years.