Get fitted for those running shoes ASAP. That should help a lot.
Switching from running on the treadmill to running outside can be difficult. The treadmill is softer, it "helps" you by pulling you along a wee bit, you don't encounter wind resistance, and unless you mess around with the incline, it stays flat. So that definitely would contribute to your soreness.
There are other things that could definitely be contributing. Per
this article, weak hips, a weak butt, weak quadriceps and tight hamstrings are all possible culprits.
When I first started running I went to see an ortho about my knee pain and she did recommend orthodics to me, which helped because my feet were a little flat. With these I ran pain-free for the next 3-4 years. Then I started doing strength training (where the person who designed the program said the most important exercise was air squats) and found I didn't need the orthodics anymore. So
stretching, foam rolling, and adding some squats to your routine might help.
But seriously...get fitted for shoes!!!