I learned to ride a bike as a 21 year old with no prior experience. Several friends of mine are looking to learn as well. I'm writing this somewhat targeted to them. Hopefully my experiences will help a bit and make the whole thing a bit less scary.
When I started out, I was pretty intimidated, but determined. It is ok to be a bit intimidated, but its important to have the attitude that you will make it happen. I think the key to learning is to keep pushing yourself slightly out of your comfort zone.
To get started out, it is certainly worth going with a patient friend to an empty (somewhat level) parking lot. I lowered the seat until I could easily touch the ground with my feet. When you get more comfortable, it is important to remember to raise up the seat. It took me a long time to be confident enough to have the seat at an appropriate height.
The goal when starting out should be gaining comfort and finding the point of balance. It is important to remember that the slower you ride, the harder it is to balance. I'd start walking the bike, then slowly trying to get a foot off of the ground and onto the pedal. For me, this was the most difficult task: having the confidence to trust the motion of the bike to carry me forward and onward. It was a great success getting both feet off the ground and onto the pedals. Once you can pedal a few revolutions, I'd recommend trying to bike in big figure 8's. That will help get a feeling for steering. Taking tight turns is frankly not super easy.
It is also more difficult to learn on a road bike, due to the way that they are set up. I found it easier to learn seated in a more upright position, and Francesca, my daily commuter is a comfort bike geared for a very upright riding position. Having beefier tires also makes it easier to learn balance. You also won't be as affected by potholes.
After you learn how to pedal comfortably for a while, and can hold a straight line fairly well, it is a good idea to get more comfortable with distances by riding on smooth quiet streets, but before you do that, read this. It is a really, really good idea to learn how to not get hit before you start riding a lot. My first ride outside of a parking lot was only a couple of miles (from Binghamton to Johnson City on quiet back roads), with my kind, patient friend, Zachary. I made an effort to ride most days after this before going on busier streets. I will post a second post about continuing to learn.
It took me a long time to get good at starting from a standstill. The best way to do this is to raise one pedal (I'm right handed, but start with my left foot, its really a matter of comfort), and push off. Starting from a stop on an upwards incline is one of the most difficult things about riding. The way to get better at this is simply to do it again and again.
This is a video from a gentleman who learned to ride at age 25. Adult learners can probably fall less than he does. Often, rather than simply put a foot down he just falls. His video illustrates the importance of learning location of brakes and controls before riding.
This is a bit more systematic approach to learning to ride. I personally don't like learning on grass, but it may work for some. Taking off the pedals can be a pain as well, as a standard 15mm wrench may or may not work to take them off.
I found both of these instructive as you can see what the process looks like. Physically it is tiring, as you are learning and getting comfortable with a totally foreign sensation.