My gears don't shift at all:
If the shifter clicks, check for a snapped cable or loose derailleur pinch bolt. If the shifter feels stuck your cable may have rusted shut or have a nasty kink and will need to be replaced.
My gears are slow to shift down for the uphills:
Cable tension may need tightening, you can try this by turning the barrel adjuster found either on the derailleur or shifter, 1/2 turn ANTI-clockwise at a time until it shifts.
My gears are slow to return when shifting into you speedy downhill gears:
1) Cable may be sticky causing a lazy shift.
2) Derailleur hanger may have been bent inwards- with the bike the standing up the derailleur should be in plane with the seat tube.
3) The chain could be dirty. A clean chain shifts a lot better.
4) Your cable tension might be too tight. Loosen by turning barrel adjuster clockwise 1/2 turn at a time until it shifts properly.
My chain falls off into spokes:
1) The most likely cause is a bent derailleur hanger which is a job for a bike shop.
2) Alternatively, the L screw needs to be tightened, this sets the lower limit of derailleur movement.
My chain makes noise when in the lowest gear on cassette (biggest cog):
1) B tension screw need to be tightened
2) Front derailleur is rubbing (very common)
3) Tooth count on rear cassette exceeds derailleurs range causing it to max out.
I'm juggling between too much and not enough cable tension to make my bike shift:
1) This is usually caused by a slightly bent derailleur hanger or derailleur cage. making fine tuning impossible. This is a job for a shop.
2) The cable is sticky and needs replacing.
3) You have a mismatch of derailleur and shifter that are not compatible. This is (hopefully) only the case if you've built the bike yourself or bought it second hand.
My gears are tuned beautifully but when I pedal hard they slip:
1) A well worn set of gears will slip under load. They will need to be replaced.
2) Your chain might be too long or have stretched over time.3) You may have a stiff link or two, check this by pedaling backwards with your hand and watching the rear derailleur for a lump in the chain.