Getting in better shape is one piece of the puzzle. A bike that fits you properly is a huge piece. Technique is also large. I see so many strong, in shape riders grinding the wrong gears at the wrong times. Once you get all of that stuff accomplished then perhaps a pair of dedicated cycling shoes for better efficiency of your new-found fitness and technique. You don't really need to train like a racer, just ride a lot.

ALL of this is on flat land:
- On cross country bike tours with loaded panniers and wide open spaces I average 13mph. I am not in a rush when on vacation.
- On city streets I try to stay at or above 20mph as I find motorists treat me more like "one of them" if I can stay above 20. I have cut across town for 7 miles at 23mph average - by far my best time ever catching mostly green lights and running a red or two.
- Twice a year I do a 150 mile day ride. My average speed is 16 mph. So I assume this is a fair guess at how I would act if I were not keeping up with cars in the city grid. 16-18mph - depending on wind direction - is very comfy for me.

Here is a short video shot from a handlebar cam that illustrates my sprint from stop in traffic. If I just made claims in words somebody here would call B.S. and I would have to post the vid anyway.