I know that every child is different and each advance at their own age. So the question is not "How old" but what should I be looking for in behaviour, maturity, coordination, and physical control to give me a good guide into knowing that it's time to try transitioning the little one from the trike to the bike (with training wheels)
Here is another option: use a 'run bike' to help your child learn balance. This makes it easier to know when they are ready.
A run bike (aka 'balance bike') is essentially a bicycle with no pedals. There is a good New York Times article about balance bikes. Unlike a tricycle, a balance bike teaches balance, which is really the limiting factor for children learning how to ride.
Once they can cruise around on a balance bike, it will be easy to see when they are ready for pedals. My son started walking his around at the age of 2, and can now run with it (at 2.25 yrs and warm weather), although he hasn't gotten the balance part yet.
There are some good questions and answers over at bicycles.stackexchange.com that include a discussion of different methods that can be used to teach a child to ride a bike.
You definitely can't go wrong with a balance bike. They put the learning in the hands (and feet) of the child. After a long struggle trying to teach my older daughter how to ride a bike that lasted 3 years, we went with a balance bike for my younger daughter. She is nearly 4 and ready to ride a pedal bike but she loves the freedom of her Kinderbike so much, we haven't pushed her to make the transition.
She's an absolute daredevil on it, riding off road, down steep hills and for some reason she really likes trying to ride the curbs. This has resulted in a few spills but the bikes are low to the ground and lightweight so even a fall is no big deal.
My wife and I love and believe in this method so much that we built an online store specializing in the bikes. There is no doubt this is the best way to teach a young child to ride a bike.
If you have an older child still trying to learn, you can remove the pedals and lower the seat to allow them to gain confidence balancing first. Find a gradual hill and let them coast down it. Once they get the balance down, put the pedals back on and watch them go.
I also think its a matter of wanting to learn. My next door neighbor kid (5) isn't all that interested while mine (4) has been wanting to for about a year...I also think the ability to listen is key...(Stop, Stay out of the street, etc).
Also be willing to do 5-10min sessions of "practice" several times a day..seems to be working so far for us.