Howdy & welcome!
Nothing wrong with your plan. Good time of year for touring. There are mountain ranges to cross, but nothing a normally fit person can't handle, and they are the most beautiful parts of the country in my opinion.
Most of the USA is safe to travel, big cities can be scary - traffic, people - but most places are fine. I'm a small woman and I tour alone, without too much worry. Picking roads is pretty important, as you can be miserable or overjoyed depending on if there are too many cars & not enough shoulder or the opposite. Once you decide your general route, you can do some research here and elsewhere to get the details figured out, and ask specific questions if you need to.
You should have no problem with town spacing. There are a few places out west where you might have to go 60-90 miles between towns, but a little planning can let you avoid that sort of thing most of the time. Mostly you'll be through a small town at least every 30-50 miles. You just have to pay attention to your maps and carry enough food & water to get you to the next town - usually less than a day.
Since you are on this site, I assume you've seen that they sell bike maps? If you choose to use one of the Adventure Cycling routes, the maps show you details along the routes of town locations, camping, places to get food/water, bike shops, and elevation profiles. If you don't use one of their routes, you'll have to figure it out yourself from other maps & resources (like state Dept. of Transportation bike maps, AAA maps, brochures you pick up along the way at the towns' Chamber of Commerce/Tourist Info places). The A.C. maps are great to get you started, once you use them for a week or two you'll get a sense of how to plan it for yourself and can more easily wander off the map's route. Most places outside of big cities you can camp for free (on wilderness / forest service land) or cheaply in campgrounds (services range from a pit toilet to full showers, store, etc). Prices are $6-$30/camp site for pay sites. In summer in National Parks you may need a camping reservation, but I rarely make reservations otherwise, a little planning day by day is enough to find camp sites.
For preparation, I would say just get your bike as soon as you get here and start riding around your new home area (lots of great riding north & south of the city) to get used to riding and get the bike fit dialed in. Buy a bike with low gears. There's endless gear shopping here too, including used gear and outlets, so I wouldn't buy anything more until you get here.
You'll also have to learn about safe food storage, since we have some creatures here that want to eat your food when you're camping - raccoons and bears mostly. Don't *worry* about bears, but do educate yourself about how to keep a clean bear-aware camp. This mostly applies in the mountains. Raccoons are at most camp sites in the west, and are really clever. Do some googling on this topic.
Welcome to the US and I hope you have a great trip! I hope you enjoy it as much as I did my tour in Victoria and Tasmania a few years ago.