Fall and winter can pose additional road hazards that can lead to flat bicycle tires. I recently managed to pick up a rear wheel flat after hitting a Sweet Gum spiked, seed pod ball. These things look exactly like the name implies- it's a ball of spikes holding the seeds of the Sweet Gum tree. The spikes puncture your tire and result in a slow flat. It generally will not blow out spectacularly. Instead, it will gradually lose air until it is down about halfway after 24 hours and completely flat after 48.
The large tear is easy to see and usually you just replace the tube. Don't be fooled into thinking you need a new, "wonder" patch or the like. The "slow flat" is tricky to find, but using the old way, you can do it competently and without spending much money. Get yourself a traditional, "patch and cement" kit. Grab some plastic tire levers as well.
The traditional method of rubber cement and patches has never really been surpassed. I've had some experience with glueless patch stickers, and they just aren't as good as the traditional method.
Remove the tire and tube from the wheel and get a basin of water.
Take your tube out of the tire and put a little air in it, enough to harden it up just a bit.
Submerge a portion of the tube in the water. Gradually turn the tube so that different parts of the tube go under the water. When you see a steady stream of bubbles start to flow up, you have found your leak.
Once you've found the leak, take some coarse sandpaper and rough up the area where the hole is. Dry the area completely when roughing it up. Deflate the tube so it is no longer hard.
Next, cut a rubber patch the size of your roughed up area and get out the rubber cement. Apply a thin layer of rubber cement on the roughed up area of the tube and let it set for a bit.
Once it starts to dry a little, remove the protective backing from the patch and apply that side apply so that the entirety of it sticks to the tube. Work out any bubbles in the patch from the center outward. Let the patch dry in place. I usually let it sit overnight.
Inspect the tire skin both inside and out for the thorn or other source of the flat. Remove it when you find it. Then put the tube back in and mount it all onto the tire as normal.