If you're anything like me, you have an easier time understanding how to do things if you understand how they work.

See the clutch in the diagram? The clutch is the part that actually connects the engine to the transmission. Mechanically, it's almost like a brake but reversed. To keep things simple, the engine side has a disk and the transmission side has a disk. When you press in the clutch pedal all the way to the floor, those disks will be totally separated and the engine can rev however you like because it's not connected to the rest of the drivetrain. As you release the clutch slowly and smoothly, you'll feel the clutch "bite": that's when the two disks make contact, but it's very light contact at first so they'll be sliding against eachother (think of gently applied brakes). The friction will still spin the transmission a little bit though, and if you're in gear the car will start to roll. As you smoothly release the clutch pedal more and more, the two disks will press into eachother more and more firmly, sliding less and less, until you've fully released the clutch pedal and the disks aren't sliding at all; the engine is totally connected to the rest of the drivetrain.

With most cars I recommend for a first step starting the car rolling on flat ground using only the clutch, no gas. With some cars you might have to give it a little gas, but in my experience most cars have enough idling torque to start from a stop by releasing the clutch very slowly and smoothly, I'm talking like 5 seconds or more to go from clutch fully to the floor and car stopped to fully released and rolling. Once you've done that a few times and you're starting to get a good feel for how the clutch behaves, you can start adding some gas to start more quickly and release the clutch quicker (but still smoothly).

The second thing to know once you're rolling is that the engine needs to run within a certain rev range, if you let it get too low (below like below idle rpms), you'll start to bog and maybe even stall, too high and you'll hit the rev limiter or possibly damage the engine if it doesn't have one. Knowing that, I think that timing shifts and knowing when to press the clutch becomes pretty intuitive, really all you need to do is press the clutch before the engine gets out of that rev range, shift up or down in the direction that gets you back into it (or maybe into neutral if you're stopping), and smoothly release. When in doubt just press the clutch, it won't hurt anything it just cuts power to the transmission.

TL;DR: The clutch connects the engine to the transmission. Press it in to disconnect the engine from the transmission, release smoothly to gradually reconnect. Start out only using the clutch to to get the car rolling, then once you've got the hang of that try adding a little gas to get rolling quicker. Shift in a way that the engine is always within it's rev range, i.e. press the clutch in and begin changing to a more appropriate gear before you start redlining or bogging.