While the returns for a long distance runner might be less for a sprinter or middle distance a proper lifting regimen can produce significant improvements. When I started lifting and built more leg and back strength I was able to significantly improvement by 400M times and have found it even helped me with my 5K's as I got older.
If you look at any professional sprinter they are jacked and that doesn't all come from running. Michael Johnson the former world record holder in 200 and 400 meters used to lift 3 days a week including Olympic lifts, snatches, cleans, deadlifts, and squats (both front and regular).
As you are just starting out I'd keep it simpler and focus on
the Deadlift and squats in addition to plyometric workouts including box jumps and round it out with things like farmer's carries and core work (Russian twist, sit-ups, etc.)
Start easy doing 2 workouts a week and then increasing to 3. You can go to the gym more often than that if you also want to work on your chest and arms but don't overdo it with the back and leg workouts.
Workout
1. Trap Bar deadlift 5x5 at 80% max (I'd start out deadlifting 145) to see what is comfortable. You can increase over time but if you are already in shape from running you should easily be able to deadlift in the low 200s and likely more but start easy and then increase weight over time.
2. Squat 5 x 10: Again you'll need to find out what is comfortable for you to lift but start out at 135 and if it's easy keep adding weight until you are able to do a 10 rep set but are getting tired on 9th and 10th rep.
3. Break things up and rest your legs a bit by doing some pull-ups, rows, and facepulls
4. Box jumps 5 sets of 10, depending on your height you should be able to do somewhere between 32-40 inches pretty easily
5. Kettlebell Farmers Carries
6. Core / flexibility work
This should all be able to be easily done in an hour
I would then alternative between
5x5 at heavy deadlift
5x10 at lighter Squat (can do front squat on lighter days)
then 5x10 at lighter deadlift
5x5 at heavier squats
For me this looks like
Deadlift 5x5 at 275
Regular Squat 5x10 at 185 or front squat at 165
Deadlift 5x10 at 205 or 215
Squat 5x5 at 225
I try to do 2 to 3 days of legs and alternate with 2/3 days of chest and arms.