There are several factors at work here. I assume you are concerned about rear shifting.

Adjustment

In simplest terms, and assuming the rear derailleur is properly mounted and the frame hanger is parallel to the cassette (not bent):

1. Put the bike in a stand or turn it upside down. Upshift the rear derailleur all the way to the hardest gear (smallest cog). Make sure you have totally "clicked out" the shifter so it is definitely all the way in highest gear and the cable is slack. Pull on the cable while you upshift to make sure it's fully extended.
2. Downshift the shifter one click. Did the cable move the derailleur? If so, good... turn the crank forward to shift the chain up to the second smallest cog. If not, rotate the cable adjuster barrel on the rear derailleur (on some bikes, it will be at the shifter or on the downtube) counter clockwise, while simultaneously turning the crank forward, until the chain goes up to the next cog.
3. Continue to rotate the crank, and turn the cable adjust barrel counter clockwise until the chain is just starting to make noise on the third smallest cog. Now dial the adjuster back clockwise until the chain no longer makes noise.
4. Rotate the crank and shift one at a time all the way down and up through the gears to make sure the shifts are precise. If not, the drivetrain may be excessively worn or the derailleur hanger may be misaligned.
   

Wear

If the basic adjustment above does not fix the problem, you may need a guru to lay hands on your bike for proper healing. It's likely your chain and/or cogs (gear cassette) are in need of replacement.

Technique

Downshifting when going uphill takes a special technique. If you downshift under full power, you are doing serious damage to your drivetrain (see Wear).

1. Do a hard half revolution of the cranks in the harder gear.
2. Back off your effort for a half revolution (use your momentum from step 1)
3. Execute the shift while your crank is just turning over, with no real torque or effort.
4. After the shift is completed, reapply power and torque to the cranks.

Think of shifting technique in terms of the manual transmission on a car or motorcycle. You wouldn't just jam through the gears without using the clutch or letting off the gas, would you? On a bike, you are the engine, clutch and synchomesh.

Chain and Chainring Wear

If you spend a lot of time on the bike, the chainrings can get worn out... as the chain stretches, the chainring teeth get worn into a pattern we call "shark tooth". Where the chain roller presses on a chainring tooth, it begins to create a depression. The depression also forms a sharp burr at the top of the tooth, and this burr will hold tight to the chain when it is under heavy load, as when climbing. The chain will resist being derailed until (as you described) the forces are greatest and the shark teeth release their victim. (Another problem caused by shark teeth is the dreaded chain suck... but that's another story.)

If your chain is stretched out and old (there's a trick to measuring the chain with a 12" ruler... ask later) and your chainring is sharktoothed, it's time for a whole new drivetrain... sad but true. New chain, new chainrings, new cogs... replacing a single item won't work, because the tolerances will be way off between the new and old parts.

Front Derailleur Problems

If your cogs and chain are OK, the problem is the front derailleur, and it's almost certainly an alignment or adjustment issue. The front deraillueur, if it's a band-clamp model, and not a braze-on model (most are band-clamps), will rotate around the seat tube if the mount is not snug, or if the bike has been laid on its side or the derailleur has met any impact.

Front Derailleur Alignment

1. Look straight down on the derailleur from above, and be sure the outer guide plate is perfectly parallel with the chainrings. If it's not, then it's time to adjust it.
2. Use the proper size wrench (generally 5mm on modern bikes) to slightly loosen the derailleur and align it properly.
3. Be sure the derailleur has proper clearance over the large chainring... generally about 3/16" or 1.75mm is good.

Cable Tension

1. Shift to low. Again, as you did with the rear, pull on the cable as you downshift so you get the cable and shifter fully extended.
2. If your bike has an adjuster for the front derailleur cable (usually on the downtube) you can now try to take up the slack in the cable until it just barely begins to swing upward.
3. If the bike does not have an adjuster, then you'll need to loosen the screw that clamps the cable to the derailleur. Be gentle and only loosen it enough to allow you to take up the slack while you slightly lift the derailleur and re-tighten the screw so the cable is under very light tension.
4. Now try shifting up to the big chainring while rotating the crank. If the chain does not go to the big chainring without skipping, raise the cable tension and try again.

Limit Screws

Check the lower and upper limit screws after the derailleur is properly aligned and tensioned.

1. Use a screwdriver to set the lower limit so that the derailleur only drops low enough to clear the chain without rubbing, when the chain is on the small front chainring and the largest (easiest) rear cog. Re-tension the cable if necessary.
   
2. Set the upper limit so that the derailleur barely clears the chain when the chain is on the big chainring and the smallest (hardest) rear cog.

Shifting Technique

Shifting the front derailleur requires a much more pronounced hard/soft pedal sequence, as described in the post on rear shifting above. Best results come from making 2 or more sprinting revolutions in the harder gear before you soft pedal during the actual shift. You should always be seated when shifting the front chainring, so plan ahead accordingly.

Set yourself up for a climb by changing to the small front chainring prior to the climb, or on a gentle upslope. Adjust the rear gearing accordingly to compensate for the lower front ratio (there are several overlapping or nearly identical gear ratios in a derailleur and cog drivetrain). You can much more easily change gears in the rear during a climb, and it's safer to downshift the rear derailleur while standing and climbing.

You don't want to miss shifts and grind gears during a club ride or a race, so practice proper shifting technique during training rides and it will become habit. Mountain biking is a great way to learn shifting, because you'll change gears much more frequently, and under heavier load, than on the road.