Keep in mind, all the time, that you need to get your 400m speed to about 4-6 sec faster than 800m pace per lap. If you're more like a sprinter, you'll likely have even a bigger difference. This doesn't mean that you do lactic anaerobic power workouts throughout the year. During the base phase, work the anaerobic alactic and glycolytic capacities. You do this by short uphill and flat sprints, 6-10sec (alactic) at relaxed top speed, and by longer, anaerobic glycolytic sprints from 100m to 200m @ 90-92% pace (~400m pace, or a bit faster). Both with long recoveries (~3-5min). This lays the basis for the anaerobic power (lactic) workouts AND races after the base. Strength training will help your speed too with the other benefits from gym work. If you do the hill sprints added with some bodyweight circuit training at home, you'll be fine, you don't have to go to the gym. 1-2 sprint sessions per week is enough.
And the same goes with the aerobic endurance. Build the aerobic capacity during the base, with some traditional aerobic power workouts (repetition training with varying distances and paces) included. But don't dig deep during base. Means avoid high lactates. Basic endurance is the basis for your aerobic capacity, slowly increase your mileage.