First off, we are talking about different issues. There is speed work that effects your maximal speed at short distances. On the same category there is acceleration. In short, the fastest you can run. It doesn't last long, and it shouldn't be trained as such.
Then theres speed endurance, which makes you run quite fast for long periods, like 25-50m or more. You also need explosive power to get your leg moving thick.
I could think of four important things to work on:
1. Lift heavy
Why? Heavy high intensity squats make you develop more force. More force means less effort on explosive action. You get faster.
2. Train for max speed and acceleration
These exercises have to be short. You shouldn't fatique too much. Take short sprints, like 15-20m. The purpouse is to train you max acceleration and top speed, not endurance. So I must emphasize that it shouldn't make you gasp for air. It's a very short sprint. You said you wanted sidesteps and other motoric skills better, this is one way to do them. Do very short sidestep sprint followed with a forward sprint for an example. Keep the distances short. This trains you to alter your movement and motoric pattern fast and hones your acceleration skills.
3. Intervals and sprints for speed endurance
These exercises are more for fatloss and anaerobic endurance. You get lactic acid (and then lactate and hydrogen). Your body learns to cope with the lactate levels and control them. It makes you long fast longer. Longer sprinting, interval, tabatha methods are all good. You could take rope skipping exercises along too. Rope skipping is also a great motoric exercise and good for coordination. Jump for x amount of time (a minute or under is preferred) as fast as possible, rest for 1.5-2 times the effort and go at it again. Running on hills and stairs is just a way to make this exercise more effective and ass whooping. It's harder on the muscles and you condition. Simple as that. Hill sprints are often recommended.
4. Plyometrics and dynamic effort
Explosive power, speed. Jumps are the easiest path. All kinds. Weighted or bodyweight, with or without boxes, every direction, with one leg or both. You name it. Dynamic effort work is doing weightlifting exercises like squats with lower weights as fast as possible. Like 2 to 5 reps for many sets. Every rep as fast as possible.
thanks for the detailed information! i think most of the exercises could be done during the season, but maybe lifting heavy is better in the offseason because of fatigue?