Choice of tire width has been a process. My needs have changed over the years. When I was young I converted the 20" useful tires to ones that were stylish sllicks. After getting a bike for longer mileage rides I was converted to skinnier tires. Then the pursuit for faster high pressure tires ensued. The pinnacle of that pursuit was my Vredestein Fortezza SEs with 160 PSI. I rode those for several years until I came upon the idea of comfort.
What a novel concept. Now instead of high pressure and narrow racing design I seek thicker wider comfort tires. I like a 26" X 1.5 or 2" now or 700C 40s. They're a joy to ride and are not that much slower for me. I also just enjoy the ride more. I ride enough on rough ground to appreciate a tire that soaks up some of that vibration.
I have never had much of a problem with flats. I've had problems with fixing flats when I've got them. Flats are best left for those who know how to fix them. To avoid flats get good tires and replace them when they get worn out. Over 49 years of riding I've had about 20 flats. I am spoiled by having someone who will pick me up if stranded. I have finally started carrying a frame pump, spare tube, tire irons and repair kit. Most of my years I carried none of these.
There are 3 valves for tubes that I'm aware of. The two common ones are Schrader and Presta. Schrader are like car tires. Presta are commonly found on 700C wheels. There are several common sizes of wheels: 26", 700C and 27". 26" wheels tend to have large tires for off road riding. They are often knobby treaded tires for mountain bikes. 700C and 27" tend to be much smaller road tires. Lately there are more fat tires available for 700C wheels, up to 40mm. Usually the pressure of 26" tires is much lower in the sub 100PSI range while the 27" and 700C are high double digits to mid-100s. There are slicks available for 26" wheels so that the ride is better for road riding. I have found this alternative appealing.