Shin pain after running
The pain you experience is likely medial tibial stress syndrome. Here is an excerpt from a study on treatment options of MTSS.
[...] However most studies support rest, ice, and analgesics in the acute phase. Many experts also recommend modifying the training routine, stretching, and strengthening the lower extremity, wearing appropriate footwear, using orthotics and manual therapy to correct biomechanical abnormalities, and gradually return to activity. ESWT, injections, and acupuncture are commonly used for other lower extremity tendinopathies with some success, but little current objective evidence exists for their role in MTSS.
In other words, the "usual treatment" of ice-packs and rest, escalating to a trip to the doctor's office if you feel the pain is too overwhelming. And if you're not already doing it, make sure to warm-up properly before, and stretch after, your exercise.
I strongly recommend learning to swim. As an exercise, it trains you, both in terms of strength, and cardiovascular endurance, with negligible wear and tear on joints, bones, and other tissue.
It's a miracle exercise, whether you're dealing with injuries or are completely fine. Depending on where you live, I'm sure there are classes you can take. I've known several people who have had to learn it in an adult age, some for exercise, some for peace of mind, and some because they got a job on a boat.