Rev matching shouldn't be required to prevent grinding on any synchronized gears; so maybe with 1st and R, but none of the others in any car produced this century.

Generally you want to upshift when engine speed reaches a threshold. The threshold can move some depending on how fast you want to accelerate, but for most driving it's probably around 2500 rpm.

When you're ready to upshift, press in the clutch pedal and at the same time release the accelerator. It takes a little practice, but you want to time it so that you don't release the accelerator too early, which would jerk you back, or release it too late, which would make the engine rev high.

After the clutch is depressed, you should be able to easily shift out of the current gear and into the next gear, with no grinding or strong resistance. If you have either grinding or strong resistance, then either you didn't press the clutch pedal all the way, or the clutch is worn and needs to be replaced.

After engaging the next gear, gradually release the clutch pedal and apply the accelerator. Again, it takes a little practice to time it well so that it isn't jerky, but it doesn't take long to get the feel for it.

For downshifting, you probably want to downshift when engine speed is around 1200 rpm. The process is almost identical. Sometimes it can be more difficult to engage the lower gear, but it really shouldn't be. Again, grinding or resistance indicates either the clutch pedal isn't pressed all the way or the clutch is worn and needs to be replaced. When the new gear is engaged for a downshift, you should increase the engine speed up to about 2500 rpm before releasing the clutch, in order to get a smoother transition. Yet again, practice makes perfect.

Another possible reason for gear grinding is if the synchronizers in the transmission are worn. That would be bad, and indicate that the transmission needs to be repaired or replaced -- or treated with kid gloves for the rest of its life.