“Shin splints” is the lay term for medial tibial stress syndrome (which sounds even worse). Long-term, probably the best way to prevent shin splints is to keep running—it’s mostly new runners or people returning to running after a long layoff who get them. If you do have frequent shin pain despite consistent running, then strengthen your arches (through walking barefoot around the house) so that your shin muscles don’t have to pick up the slack for them when you run.
If you’re having a bout of shin pain, run on soft surfaces if possible, and avoid downhills to lessen the pounding. Try to keep running, albeit less than usual if necessary, to keep the consistency that should eventually help you leave shin splints behind.