I’m not seeing my walking and running pace change, what should I do? Also I don’t have Apple Watch is it the reason?

You need an Apple Watch to use most of the Fitness app features. However, with iOS 16, see:

Track daily activity in Fitness on iPhone - Apple Support

"You can use the Fitness app  to track your daily activity, set a move goal, see your progress, and see your movement trends over time—even if you don’t have an Apple Watch." I believe that is all you can do without an Apple Watch.