I've been driving with clutch+6-speed since gt5 came out. Will concur with the poster's above, it's the game, not the wheel. N=neutral, for reference.
Upshifting:
1) Throttle must be fully released (except from a standstill in neutral, then you can launch at any throttle in any gear)
2) Clutch must be fully depressed before the stick moves from N to higher gear.
2.1) Count out a "1 mississippi" when moving from 1st to 2nd gear. Just like in real life, the big ratio jump requires a bit more time in N for the engine revs to drop for a smooth shift. All cars in game do this, it's double to triple the time of other gears, and the cause of many missed shifts for new players.
(It's fine if you're still mid-stroke when pulling from low gear to N, but it must be depressed for N to higher gear. Yes, you can "lazy man double clutch" by popping into neutral, rev matching, clutching, and shifting. Personally, it's something I just happen to do IRL anyway.)
-Fastest way of doing this (in my experience) is to begin pushing the clutch before letting off the throttle. Drop into N as you let go of the throttle. You should have the clutch fully depressed by now, click into a higher gear, let go of clutch, re-apply throttle.
-If you miss an upshift: STOP. Don't panic, hitting pedals and moving the shifter until you get a gear.
-If you're in something that matters (tt, race, etc), just click the paddle. It'll move you up, you can clutch/shift the next gear change and be back to normal.
-If it doesn't matter: Let go of the throttle 100%! Clutch in 100% Shift down a gear, back through neutral, and into the right gear. Release clutch, apply throttle, drive off.
-Keep in mind, the few hundredths of a second you may save by "shifting quickly" is destroyed by one missed shift. Shifting consistently, lap after lap, is always going to be faster over the course of a race.
-If you need the speed, the triple plate clutch noticeably speeds up the time the game will allow you to do this process. This is ESPECIALLY true on cars with big ratio gaps (M3 csl, for instance) and cars with big lazy motors. I put it on all my cars, just in case.
Downshifting:
1) Right foot off throttle, on brake.
2) Holding brake, use left foot to clutch all the way in. You'll hear your RPM's drop as soon as the clutch disengages.
3.1) Did I mention CLUTCH ALL THE WAY IN? That's where most people screw up, as they brake 50-80% and the clutch just seems to go to the same point naturally in most people.
3) Pop into the next gear down, release clutch. You'll hear the engine RPM pick up again. Beware of the increased "braking" power of the drive wheels as the added resistance of the motor is added-it might cause a slide.
3.1) If you are rev-matching/heel-toe'ing, this is where it happens. Pop the gear down, clutch still in, rev match and release clutch. DO NOT rev match in neutral, or in the higher gear, as it will cause a mis-shift.
3.2) You're supposed to hit every gear on the way down. 6th gear straight to a second gear turn (think Tokyo or Fuji) means you're hitting 5th, 4th, 3rd, and finally 2nd while braking, clutching between each shift.
4) Repeat as necessary. It'll be second nature in no time.
If you miss a downshift, no biggie. Keep braking, clutch in, and downshift to the next gear like you were going to anyway. The effect on your lap time is negligible unless you panic. No faster way of doing this is needed, the shifting should all happen within the braking period with time to spare, and the exact timing of the gear changes produces a very small difference in lap times.
When I was teaching my friends, we were all saying "throttle off, clutch on, shift gears, clutch off, throttle on" every time they mis-shifted as they were learning 6-speed.
I'm not some GT elite, but I have no problem taking any car around any track without missing shifts. I routinely do 5 lap races without a single miss.