I taught my 4 year old son to ride without training wheels in about 3 weeks, while I was at work and not even home! This was after it took my daughter until she was seven to learn the same thing. I didn't want him to take as long so I went looking for tips, and read how someone else did it. It does work!
Step 1: Recomended (strongly!)
To do this you need a child who can pedal, and who rides a bike with at least 12.5" tires and a coaster brake or hand brake.
Your kid should be pedaling and braking well,
They should wear a helmet
they need to be under voice control (stop when you say "stop")
they need a flag on thier bike also, as they only come up to the top of the hood on most cars while riding
putting a bell or horn on the handle bars gives them (and you) a quick way to give turning directions as at this age," right "and "left" may not make imediate sense. "toward the horn side" or away from the horn" will make much more sense
they need to wear sneakers or sandals that firmly strap on, flip flops come off easily and let toes get nailed by stuff on the ground. barefoot allows toes and even the foot to get crunched or cut easily.
Step 2: Look Ma! No Feet!
Before you do this step you will want your child under voice control ,as once they learn how to ride without training wheels they can get out of reach really fast!
You want to lower the bike seat until the child can place both feet flat on the ground while straddling the seat, until they can do this they are to small.
Now take the training wheels and pedals off the bike, REMEMBER the pedals are threaded so that they tend to tighten under normal pedaling motion, this means one is threaded backwards! (left hand thread) turn the pedals forward a few times until you figure out the correct loosening direction for each one.
I show my adjustable wrench removing the pedals, if you can, use a regular wrench as it will tend to strip the nut less and have better access to the nut due to having a smaller head.
If the pedal won't unscrew:
-make sure you're turning it the correct way
-soak the exposed end of the thread with coke, WD-40, liquid wrench, or any other penetrant oil,
-try a torch as a last resort, but don't melt the pedal or pedal bearings, or set the bike on fire.
Step 3: I'm Ready Now!
Let your kid "run " the bike around the yard until you think they are ready. I was at the kittery trading post (like LLBean, or cabella's, but better and better value (I think)) in the bike department they sell bikes with no pedals or chain, to do the same thing, and best of all you can pay $70 dollars for a "runner" that your child will out grow in a few weeks.
If you have a small hill in your yard have the child push the bike up the hill and ride it back down. Make sure they can stop before hitting anything by dragging their feet, as they don't have brakes!
Once you see that the child is not putting thier feet down until they are going to fall over because they have almost stopped, it is time to put the pedals back on.
My son started complaining that he was falling over due to not being able to pedal after about 3 weeks, I watched him and found he could have kept going without touching the ground if he could pedal. He was instinctivly turning the handle bars and leaning to keep his balance.
Step 4: Look MA! No Hands!
After you are sure your child has learned thier balance, it is time to put the pedals back on. I lucked out on this as I had two 12 1/2" bikes (both for free that others had out for the trash) the one I pulled the pedals off of has a loose pedal set bearing anyway so I gave it to another kid who wants to ride without training wheels.
Install the pedals tightly so they won't back off under the stress of pedaling, I let my son ride his bike with the pedals on it in the yard first so he could get used to how to "launch" from a standing start. He was afraid to ride it on the street for a few days even though he was riding in the yard, but after I ran holding him once or twice, he learned he could go fine without me.
This is the important step for parents; ride with your kid as much a possible for a few days so they get used to how it is done. Compared to my 21 speed and my daughters 18" bike, he looked like he was running on the 12 1/2" bike to still go slow. I tried him out on the next (free) bike I have, a 16", and he was ready for it. The larger tire sizes have longer frames and wheel bases and make for a much more stable ride than the smaller tires do.
My wife walks 3 miles everday, and part of her walk is up a large, long hill. on his training wheels my son needed to be pulled (my wife has a rope on her jog stroller to pull him, but doesn't need it any more) just like my daughter did until she learned also. Coming down the hill he would be scared at the speed he would hit on the 12 1/2" bike, on his 16" bike he isn't. When he tried out the 16" bike we went for a spin around the block, my son had trouble using the brakes, as the crank arm is longer and the coaster brake requires a longer movement to engage, we then spent a few minutes with him learning how to stop and start.
Make sure your kid knows they need a helmet to ride, that they know not to ride in the street unless it is safe (your call).
Bikes that don't have front, rear, wheel, and pedal reflectors aren't realy meant for on the road (especialy at night), and if you ride at night you need lights. Until he child has all the reflectors (my daughters 18" bike does) my son rides on a tag-a-long bike with me on roads without margins or sidewalks.
If the bike doesn't have fenders, you may want to put some on as all kids love to ride through puddles, and will end up with a streak on thier back!