I am still pretty slow compared to many, but my speed has increased over the last two years to where I am consistently close to 16 mph. The first summer I rode I was at 12-13, the next summer 13-15 and now starting my third year of riding I'm sometimes breaking 16. (I am 49, btw)
I did not do a training program or intervals. All I did was ride with a group who, fortunately, was at the same level I was, but we did push each other to go a little faster than we would have on our own. None of us have super duper bikes. We are mostly entry level and mid range. I did upgrade the wheels and tires on my bike with a good used set I was lucky enough to come across, and I believe that did help me a little.
I think the biggest boost was that this past winter we all did spin classes and rode outside when we could. I think that was zero times in Dec., once or twice in Jan. and Feb., but more the rest of the months. That meant we were not starting over in our fitness level in the spring and could build on what we had worked up to the summer before. That really helped.
So my advice: Be patient with yourself and have fun at the speed you are at right now! Ride as much as you can. If possible, find people to ride with who will push you a little. I have been amazed at the people who come up to my group at stop signs and stores where we stop for a break wanting someone to ride with! There are lots of people starting out. Keep fit over the winter with a your bike on a trainer or spin classes and as much outdoor riding as your climate will allow.
Grits
2010 Trek 5.2 Madone WSD, SI Diva Gel Flow
2002 Terry Classic, Terry Liberator