Did you check the inside of the tire? The rim strip? If you mount the tire brand logo centered with the valve stem, you can find the hole in the tube and then quickly determine where the flat came from on the tire in relation to the stem and logo. Always inspect the inside of the tire too.
It's a good idea to clean your tire "carefully" with a gloved hand while you ride if you can, Practice this a few times, if you're unfamiliar, and certainly inspect your tires AFTER every ride. If you wait until before your next ride, you'll keep your freinds waiting, but check the pressure before. Some tubes, especially latex, lose air pretty quickly, and temperature (Charle's Law) can effect them too.
Tubes aren't going to pop out of most small cuts. If the cut goes through the casing, replace it. You can use a slit section of old tube, an energy bar wrapper, or a patch kit to re-inforce a tire area that has a cut or gouge that doesn't go through, or to get you home on one that does.
Make sure you inflate to proper pressure to avoid the pinch, and check around the tire bead before inflating for any tube caught between the rim and tire. Try not to use your tire lever to re-install the tire if possible, although it can be really tough.