As you have discovered there are 2 types of valves in common use on bicycles today. The Schraeder valve is the larger one (a little over 0.3" in diameter) found in automotive use and on older, cheaper, and more utilitarian bicycles. The Presta valve is narrower (about 0.2" in diameter) and is common on newer, dearer, and more "serious" bicycles. While the Schraeder is held closed by a spring that may be opened by pressing central pin, the Presta must be opened or shut by turning a knurled nut on a central pin. The two are not compatible although most good floor (home) pumps have a combination chuck, most frame or mini (carry on the bike) pumps can be set up for one or the other, and most CO2 cartridge inflators are Presta only. There are screw-on adapters that allow one type of pump to inflate the other type of tube; these are about the size of and as easy to lose as Schraeder valve caps. There are also grommets that reduce Schrader sized rim holes to Presta size to prevent the smaller stem from moving in the hole and causing the tube to fail.
The most common and expedient method of dealing with a flat on the road is to replace the old tube with a new one after checking for and resolving any obvious cause. Obviously you or a companion must be carrying a spare tube and a pump or other means of inflation. If you cannot make the repair or there is some urgency such as a time commitment later that day, the next method is a cell phone call for a lift. Depending upon the likelihood of an incident and the consequences of not having tools and spares you have to decide what to carry.
Once home I throw the flat tube into a box of tubes to be repaired and take a sound tube from a box of new/repaired tubes to put on the bike. When I have 5 or 6 tubes that need repair I will repair them. Bike shops sell repair kits that generally include a few patches, a scraper or bit of sandpaper, and a tube of rubber cement. Be warned, the tube of cement is good for one use. Once opened the cement hardens and will be unusable the next time you need it. Save the tube for emergency use. I buy an 8 oz, "brush in screw cap" can of cold vulcanizing cement from an auto parts store, Walmart, or Amazon. It lasts about 5 years before enough volatiles have evaporated to make its use questionable.
Most of the rest of my "home" tube repair kit is assembled from "Rema Tip-Top" brand components available through bicycle shops or on the internet. Rema items include 100 piece box of size F0 patches (5/8" diameter for 23-25 mm racing tubes), another box of Size F1 patches (1" diameter for larger bike tubes), a box of Size F2 patches (1"x2" oval for wheelbarrow, lawnmower, etc. tubes), and a 1 qt can of buffing fluid (now about 20 years old and half used). Other items include a brass bristle brush, a "stitcher" (a serrated edged roller), and a floor pump.
The home repair drill is a follows: 1. Find and mark the leak. Some small leaks are so slow that under water they release a tiny bubble a second as opposed to a stream of bubbles. I will usually enlarge the leak with an awl so that the hole will "telegraph" through the cement. 2. Buff the area with buffing fluid and the brass bristle brush. This will usually remove the chalk or ink that I marked the hole with, hence the use of the awl. 3. Apply the cement and wait, usually at least 15 minutes, sometimes I have forgotten for an hour or more. 4. Place the patch and stitch it down. 5. Wait, typically overnight to ensure the cement cured. 6. Pump up the tube to see that it holds air (typically overnight). 7. Deflate the tube, put a cap on it so the center pin (Presta) or edge of the stem (Schraeder) will not wear a hole owing to vibration while being carried on the bike, and roll the tube securing it with a rubber band. 8. Put the repaired tube in the box of sound tubes. Five or more tubes at a time is more efficient that one or two at a time.
