I started to go for light runs for the first time in years. I can now manage a 10.6 km/hr pace over 15 minutes. However, after 15 minutes or so my shins start to get a little sore. They still feel sore and ache a little bit 1hr later.
So what do I need to do? Also, what is a reasonably good pace/time to shoot for? I am 5'11" 155lbs.
Walk more, try to increase your weekly mileage by a lot.
Do this for 6 weeks.
Start running again with very low mileage e.g. 2x a week for 10-15min.
Increase frequency after six weeks to 3x a week.
After six weeks again to 4x a week.
After six weeks you can increase the duration to 20min.
Six weeks later you can go to 30min.
This is a super slow plan. But as the saying goes, people overestimate what they can achieve in a year and underestimate what they can achieve in five.
The six weeks period is long but it allows for full adaption of the tendons which is often the slowest to adapt to increased load.
I went from sedentary to running 70K a week in 1.5 year. My times on the 10K dropped to below 40'. When I started out, I think that I would do +50min on the 10K.
The slow increase in load helped me to never get injured. While the schedule above goes slowly, it becomes 'fast' in the long run because it helps avoiding injuries.
As for a time to aspire. I would say that sustaining that pace over 10K could be a first (long term) goal.
Good luck.