Get longer runs in and increase frequency, perhaps to 4x a week. Keep increases gradual to prevent injury.
May not feel like it but you will eventually get faster.
Have to warn you, however...once you start running longer and you get more hooked you will want to keep increasing mileage.
When I started I just wanted to get out for 30 minutes of running to get cardio up and lose some weight. Now, 2 years later, I can feel lazy if I don't go out for at least 6 miles.
Regarding speed training, I will eventually get to it once I really start to plateau. I'm still seeing improvements just by getting the distance in.