I have done solo camping both on foot and in a vehicle
If you do it on foot, you might be inclined to hike one of the major trail systems in the US -- sleeping anywhere in between.
In a vehicle, you can park at virtually any walmart overnight.
A sleeping bag more than suffices to keep you warm at night -- just make sure it's the proper rating for the season.
The big challenge is food. The best thing to carry is freeze dried meals, but those tend to be expensive. But they are the most lightweight and all you need to do is add hot water. (In a car, weight doesn't matter as much, of course)
For hot water, you need a butane/propane tank with a stovetop and pot (but don't keep the tank inside a vehicle at night lol).
Don't forget a portable water filter, and external battery pack for your phone.
A sealable "travel bag" is ideal for storing your clothes, to protect them from the rain. It can fit inside a hiking pack.
And don't forget about bug spray and similar precautions. Tics are a major issue and can give you lyme disease (I've been bit multiple times and gotten infected at least once). Spraying your clothes and tent with permethrin, and making sure your tent is fully sealed at the end of the day should keep you safe.
For the most comfort, you can invest in super lightweight equipment to ensure that you aren't carrying too heavy of a load. Lightweight gear tends to be more expensive, though (e.g. lightweight tents). A tarp is extremely useful against the rain, even if you have a tent already (the rain tends to get through).
And finally, having a pocket knife and pepper spray will be handy in case you need to use them
