Disclaimer: This isn't an authoritative "guide" just my own meandering thoughts on how to get used to MT mode.
To learn how to get the most of MT mode, I'd suggest that you take a moderate car with easy handling and power (RX7 or similar) to route-X (arcade time trial) to start, since it has only two curves. This initiation will help you to understand the dynamics of the acceleration deceleration it brings you. Make sure to apply sports med/soft for the first test. Good grip, but not to the point of true stickies.
** do not attempt to *learn the MT using races. let that wait for a while. you have plenty to do before racing. (ok, you can do as you see fit, but that's my recommendation)
So now, practice increasing and decreasing your speed using only the gears. Shift up when the shift indicator tells you to. take it up to your max speed, and without using brakes, allow the car to slow to ~5500 rpm, then shift down, allow it it slows more naturally (till around 5500 rpm), then shift down, etc. until you're almost stopped in 1st.
Do this over and over and over again. Experiment with shift points (shift a little late, shift early... see feel hear the difference)... *listen for the whine of the motor. It tells you when to shift. Even the best of us will probably glance at the shift indicator once and a while, but I think I can speak for others who prefer MT by saying that eventually, you'll hear when to shift. Sometimes it corresponds to the shift indicator sometimes a bit late or early. (shifting ahead or late can benefit you in many ways, depending on the situation at the track, providing transmission braking or a bit of extra kick.)
So once you're really comfortable shifting in this simple situation, take it to the next step. Go visit the high speed ring in an arcade timetrial.. It has a few curves to help you start to feel the MT during these accel/decel events.
Very important: Drive slowly at first, try not to break ~70-90mph (~110-150 Kph equiv). Spend a few dozen laps improving your lap time.
Now, like at route-X, I'd like you to try to avoid using your brakes expect when ***absolutely necessary. You're not going all that fast here, and in most cases (except the S after the bridge), you should be able to do all of your braking using only downshifts, well placed of course. And this is exactly what this particular exercise may help you to learn.
Use a combo of coasting and downshifting. Coasting (or the obvious braking we're avoiding right now) is important when you've got your tach pegged near or at the redline. the gears will shift far better when the tach is well below the redline. try to shoot for ~1Krpm difference (ie, redline ~8K, downshift once below 7K..., etc. all depends on the car you're using) or a bit more to downshift at first, remembering that once you've "got it", and except for the redline itself, the specific value and difference will be meaningless (since you'll shift when you need to without looking for the most part, and the shift points will be almost hardwired into your muscles )
When you feel comfortable, exit the time trial. Yes., exit the time trial. Go get a drink of water, "relieve thyself", eat a snack, etc...just do something else... In other words, wait a little bit, just walk way from the console. There's a purpose to my madness, which has something to do with muscle memory and concentration
So now that you've done your mandatory relaxation period, let's get back out onto the track. Go back to the High Speed Ring. Crank up the top speed. Not full balls out, but moderately fast. The RX7 (for example) has a practical top speed of about 170mph or so (quibble quibble snark snark snark). Keep yourself to around 150mph max. You're still going to rely on the shift indicator, but if you "hear it" well then, shift. In fact, try to use the shift indicator as much as possible, but again, if you "hear it" at least consider shifting. Remember to coast just a little before downshifting.
Round and orund you go, Push your lap time. Forgt the ,max of 150. Push it more, try to use your shift points to help slow you into cruves. Most times shifting down will help, sometimes it'll cause you to spin. Sometimes shifting up for a sec can help to stabilize you. Experiment. Spend the "day" doing this.
Once and a while, just get up and walk away for a little while, maybe 5-10 minutes.
Graduate from the high speed ring to another venue. And not that you should leap to it, but I personally like NurburgRing (not the race course, the road), but you decide. Maybe it's time for a quick race in the lower classes (novice, A or B) just to see how you're doing. Do it!
If you find you're unable to keep up, switch back to arcade mode and keep practicing with the basics. It'll happen, (and when it does, you'll suddenly find yourself in more control) although it may take longer or shorter for some. Good luck.
PS> Best use with wheel, although I was using MT and a DS-3 within a month of buying GT5, (didn't get a wheel for almost a year) my first racing game, so it's not like you can't.