How should I handle soreness in my legs after running? What additional tips do you have?

Something you'll learn is that most adults have terrible running form. I was running for 10 years before I learned that I was an under pronator and that caused me constant IT band injuries and rolled ankles. I focused on my actual strike path and things improved. Later I tried chi running and pose running and I found chi running let me run forever without pain.
Second thing is shoes. Depending if you're a heel striker or midfoot or toe striker will determine what the drop on your shoe should be. Midfoot strikers use a zero drop and heel strikers usually use somewhere between 4 to 12 mm. Having more cushioning in the sole helps as well. Don't wear a running shoe longer than it's life cycle and never lift in your running shoes or run in your lifting shoes. Putting excess weight on your shoe destroys the midsole cushioning.
For recovery on days like this, marathoners swear by ice baths and cold showers. It brings down the inflammation. You might not notice the effects immediately, but it can cut a day off your recovery.

Keep walking at a slow comfortable pace after your run, breath deeply as you walk, if you always do that after your runs, sooner of later your body will get used to the running, and your legs won't be sore anymore.

I love running, but it takes time to get use to it. So, start running no more than 20 minutes every other day for a few weeks. Don’t worry about your speed right now. As weeks go by, you can try to run on a treadmill to work on your pace and timing. Drink plenty of water before and after your workouts.