basic premise is:
High reps with low weights builds endurance
Low reps with high weights builds strength

though it's much more complicated than that.

DOMs is just usually a sign of doing too much too soon, and could be caused with either sort of training.

I'd go for squats, single leg work once strong enough, dead lifts, RDLs, lunges (forwards, back, sideways) and some plyometrics such as hurdle jumps and box jumps plus step-ups and weighted steps ups.

2 exercises that are perfect for runners are get-ups (search turkish get-ups) and burpees.