Hi there - sorry to hear about your calf/shin pain. This is very common in runners, especially new runners. I can't comment on your running background and history as you haven't mentioned it. I also can't really diagnose this for you. I would always recommend you see a professional physio asap if you are at all concerned. Best advice is to find a good local physio who you can develop a good relationship with over time. They then get to know your body and can help you get to the bottom of just about all issues.
That said, I have had a lot of calf/shin issues in my almost 10yrs of running. So a good understanding of what can be going on is useful. Basically with calf/shin you need to achieve a good strength balance between the shin muscles at the front of the leg (particularly
tibialis anterior) and the large
calf muscles at the back. There are many strengthening exercises that you can do so I won't go into them here, suffice to say there are loads of resources online and on YouTube.
Another important muscle you should be aware of (and I think this may be the source of your issue) is the
tibialis posterior. It has a tendon that runs around the ankle and along the tibia. Any inflammation or dysfunction with this muscle can cause issues. So again rehab and strengthening of this muscle should help as well.
A good approach (outside of seeing a physio) would be -
1. Rest and icing of painful areas;
2. Massage of key muscles (if not painful);
3. Strengthening protocol of key muscles (if not painful);
4. Gradual return to running being careful not to try and jump back in where you left off. Slowly but surely is the best way, backing off if any symptoms re-emerge.
Hope useful. Very short (1min) video
here on basics of shin splints and pointing out those key muscles mentioned above.