I am a master of stealth camping, I say with all due humility.
Look at your map and plan to be in a lightly populated area around dusk. Look for the green on the map; forested and woodsy areas, national parks, or farmland. Hilly country is better.
Plan to set up camp with the day's last light, and get well off the road when no one is looking. Try to find a place where there is no reason for anyone to come around. Get practiced at making and breaking your camp quickly. Try to be at least a few hundred yards from any well used road.
Some good tips:
Look for orchards where you can tuck away behind rows of trees for the night. Also river bottoms, side roads in the desert, logging roads, railroad right of ways, roads that serve power lines, fire roads that lead away from built up areas. Farm roads can be good too, but areas with livestock are likely to be fenced in and watched more closely. If you can follow a farm road to an area that's not being farmed, that works. On the coast, beaches are usually a bad choice unless they are really hard to get to. People party on beaches, and the authorities usually patrol them. You want to be where nobody has reason to be.
You may be able to camp behind the church in small towns; ask nicely or just leave a small footprint and be on your way early.
Try to avoid places where teenagers and drunks like to party. Lots of beer cans...look elsewhere.
When you want to visit a city and spend a day or two, try the hostels. Look up AYH and get a membership. Cheap dorm beds usually, and a chance to meet fellow travelers and share stories. If you want to stay a few days at a hostel, ask the manager if you can work for your bed, pulling weeds or cleaning up around the place. Try warm showers lists and couch surfing sites as well.
Plan on two meals a day from the supermarket, maybe one at a restaurant for a bit of local flavor and comfort. I always have a quart of water, peanut butter, bread, and some fruit with me. Can always make a meal, even if its unplanned. I don't do much cooking when stealth camping, but oatmeal and pasta dishes are cheap and quick.
If you must stay in campgrounds, ask the host if they have simpler sites like a hiker/biker site or a group camping area. Explain that you need a tent spot only, just for the night, and are on a budget. No need for hookups etc. Plea poverty and offer them five or ten bucks. They may work with you. Campgrounds that cater mostly to the RV crowd are not promising in this regard.
I have been known to pitch my tent on the "periphery" of a campground, out of sight and out of mind. Unethical? Well, I am no saint. Do a good deed some day and balance it out.
If you do get caught where you're not supposed to be, try to convey that you're on a shoestring budget, having an adventure, and that you mean no harm...and you can be on your way in a jiffy. Smile a lot and be nice.
You may be able to score a shower at a municipal swimming pool, or a campground where you're just passing through. Be discrete- park your bike elsewhere, like in a picnic area, pocket a bar of soap and use a tiny little towel you can tuck away. Or just offer to pay for a shower. You can often pay for a shower at a YMCA, and have a workout to boot.
Good luck and good travels.