When you exercise try to stay in heartrate zone 2/3. This may mean walking up a hill as many people cannot start running and keep their heart rate down. Many who try to run get discouraged as they go too hard and blow up their heartrate which makes for a unpleasant experience.

Over time the speed and duration you can run will get better but your heart rate will stay the same.

I would recommend trail running as it is much more dynamic and you are less likely to get overuse injuries like people who run on concrete for many miles get stress fractures. Bonus points you get out in nature.

Our heart rate will not stay the same when we run.

Our expectations change. We learn to expect a faster heart rate.