I'm not a physio-wizard, but have been around long enough to offer one or two bits of advice:

First, don't worry.  If you are puting in the effort on a regular basis you will be healthier each day as a result  and will improve in time.  See if you can find the joy in movement that running can bring.

Second, don't pay too much attention to distance or speed.  Try to work out by time instead, and increase your time gradually.

If you want to take it up a notch or two, you can invest in a heart rate monitor and read up on how to maximize the aerobic value of your workouts.  I don't do this stuff myself, being more of the "just go and run" school.  I have been reading posts by people who have achieved great improvements in their health and performance using the HRM based methods.

Good luck!  Have fun!

I am not sure why you are unable to get over the 3-mile distance after 5 months.  Maybe there is a particular route or specific routine that might be blocking/interferring with continued improvement?    I would only suggest to try a walk/jog/run workout to go ahead and achieve putting in the additional miles.  Regarding the 5k goals, try scheduling one day a week for speed workouts.  (As fast as you can, with rest periods and then increasing speed again, eventually pushing out the length or sprints and overall distance).  This will help with establishing/controlling pace for events.  Still, for endurance, it is important to break move beyond the three miles, so use any combination necessary to get beyond.

Wishing you the best!

JimmyB_42 Great info on those links. Thanks for sharing. I think in our quest for speed we sometimes overlook the basics.

To work on my speedy I would do interval training on the treadmill.  Run for 1 min, jog for 3-4 min, run for 1 min and then back to a jog for 3-4 min.  Do this over and over again till you've reached the time or distance you've set on the treadmill.

Hope this helps.  Good luck.