Roadie for close to 40 years, worked many years as a bike mechanic.
This says rim strips all over. Punctures are relatively slow.
Go to your local bike shop and buy 2 rolls of cloth rim tape and 2 good inner tubes. Like Continental or Michelin. Thinner is better - it will conform around stuff better.
Back home, remove both tires and assuming they're not wire bead, turn them inside out, carefully inspect for debris. The cotton ball idea is great - we just used our fingers and paid w/ The Bike Mechanic's Badge: one continuous thin laceration perfectly in line across all 4 fingers of one hand. You may need fine tweezers to get a little speck of glass out. Look at the outside too - I had a bit of a thorn once that wasn't visible inside or out, but under load it would poke the tube eventually.
Tires are good? Now remove the old rim tape or strips, clean the valley of the rim, check for burrs at the rim joint and dress them if necessary. Wipe down w/ lacquer thinner, let dry, and apply new rim tape.
Now - and this is important - put just enough air in your new tubes that they hold their shape and ~generously~ coat them in talcum powder. Tubes used to come well talc'd, but not anymore. Also talc the rim valley and the inside of the tire.
In case you've never been shown, there's a right way to mount a bicycle tire:
Place one bead of the tire over the edge of the rim, and use your fingers to work it on. As it takes no additional time, put the tire label on the drive side of the bike at the valve stem hole. Now when you go to air your tires it's easy to find the valve stem.
Then stick the valve stem in the hole and work the tube into the body of the tire all the way around. Then, starting at the valve stem, move the tube over so it's in the valley of the rim.
Finally, starting at the valve stem - with the side of the wheel with the tire bead not yet mounted facing up, away from you - using both thumbs and working simultaneously away from the stem, work the tire bead over the edge of the rim. Some super-light race tires will require letting out the little bit of air in the tube before the final snap.
USE YOUR THUMBS ONLY! A bike mechanic will use a plastic tire lever to get a bead off, but never on.
The beads opposite where you're snapping the beads on will be down in the valley of the rim to give you slack.
Finally squeeze the beads together and wiggle to make sure the tube is free and centered in the valley and not pinched.
Air it up, ride your bike, and you won't get flats.