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This was something I figured out back in high school. I had a beater I didn't really care about and I had a bruised left ankle, so I hated pushing the clutch in. I knew conceptually that I could shift gears without the clutch, and so I took the opportunity to learn this until I was able to pull it off just about every time. I use this trick everyday, because I am lazy. Note that this can be a good way to ruin a transmission and some cars have lockouts to prevent this sort of destruction. This is an intermediate to advanced manual transmission topic.

The concept behind this is negating the force of the engine with respect to the speed of the transmission. While the engine and transmission are always matching speeds while the car is in gear, the engine is usually putting force on the gear i.e. while accelerating and decelarating. At the point where the engine is not putting force on the gears is the point where a gearshift can move freely out of gear. When the engine speed is matching the transmission speed (speed of the gear) is when you can freely put the car into gear.

This can be done for either upshifting or downshifting.

While in gear, accelerate to the point where you would normally shift. Now ease off of the gas while putting about half the pressure you would normally use to shift gears. If the gear pops out you're halfway there, you've shifted out of gear without using the clutch. If it didn't work, try again, maybe you've taken your foot off the gas too quickly. Maybe try a different gear. It's better if the engine RPM's are lower as there won't be as much of a jolt to the transmission, especially if you miss. But the basic idea is Less Gas and Pull. This should be one fluid motion

OK, now to get back the car back into gear, bring your foot off of the gas pedal as usual, and gently push the lever into gear. At some point with the engine speed decreasing it will match what it should be exactly. If you miss give the car a little gas while still applying pressure to the gearshift. If you're engine idle speed is too high you may have problems, or if you're going up a hill (living in New Orleans so long I've forgotten that people encounter hills on an everyday basis)

It's easier to do this in a place where there is not a lot of traffic, when you're not in a hurry, and higher gears are more forgiving about this. Try pulling the car out of high gear next time you're on a desolate highway without using the clutch.

Special note on first gear: this trick will not get you from a stopped position, but you still can shift into first gear without the clutch. This is how I stop for the numerous four way stop signs on my way to my house, providing of course that I can see that nobody is coming. Using the brakes with the car in neutral, I slow down to about 5 MPH or so, and I push the gearshift into first. When it goes in, and I want to go again, I just push the gas and dispense with any of that clutch pushing dreck. If I need to stop I of course push the brake and the clutch as usual.

Guess what? This is a real world application of calculus!