I spent several hours looking this stuff up online a few weeks ago. A few takeaways I learned were that (according to the research):
1. Runners should lift weights twice a week, specifically on their hard days. So if Tuesday is track work and Friday is tempo, you lift those days.
2. Weight lifting should be done at least 3 hours after finishing the run. So run in the morning and lift in late afternoon or early evening would be the norm.
3. Warn up and cool down for 5-10 minutes of easy cardio, preferably non-impact.
4. Use 3 sets of 5-10 reps per exercise with about 60 seconds rest. Goal is to lift heavy.
5. Runners who consistently lift twice a week reduce injuries by one third. So, for example 3 injuries over a certain time period without lifting weights would have only been 2 injuries with weight lifting.
6. Lifting can improve running economy by up to 8%, but most will likely be in the middle (3-5%).
7. Try to prioritize compound exercises.
8. Upper body lifts prioritize chest, back, and shoulders for better running posture.
9. Routine should start with 1 squat, 1 hip hinge, 1 lunge, 1 push for chest, 1 pull for back, and 1 shoulder exercise. Over time, it's okay to expand the routine and do more than that, but it should at least be comprised of that.
10. During the taper, you taper weight training as well. So the routine stays the same, but instead of lifting twice a week, you only lift once a week.