The most immediate ways to increase your speed on a bike are not training exercises that improve your fitness. They are skills.
Don't use your brakes
Seriously, they only slow you down!
OK, that one is mostly a joke; but it is also true. The more you can avoid using them, the faster you will be going given the same amount of effort.
Ride in a Paceline
By riding in someone else's slipstream, you can really decrease the amount of energy you expend. It really is amazing. I've heard people say you save 20 or even 30% of the energy by drafting. I haven't seen any mathematical proof of those numbers, but they seem pretty reasonable to me. The first 3 minutes of this video explains how to do it safely.
Ride Over Hills rather than To the Top
A lot of riders focus on getting to the top of a hill, and recover before accelerating. If you accelerate when you get to the top of the hill, you can recover while going at your target pace, rather than recovering at a slow pace.
Get aerodynamic as often as you can. It might look silly; but it works.
Stop Pedaling Downhill
I find that I often I can go faster downhill if I tuck and don't pedal. It also saves energy.
Stay in the Saddle and Spin
Stomping on the pedals is an extremely inefficient way of moving a bike. Instead of climbing out of the saddle, go at whatever pace you can without going anaerobic. Remain seated and spin.
Use it at the End
You've done a lot of things to conserve energy through your ride. If you want a fast pace, at some point the value of staying aerobic is gone. Use every last bit of energy and really push to the end.
I love recording a personal best pace; but I also ride on the roads. It doesn't make good sense to shave time by doing stupid things like crossing intersections dangerously. Stay safe.