Good afternoon prynsys4NYC,
Welcome to the Apple Support Communities!
I understand you are not seeing the flights of stairs you are taking register in the Health app on your iPhone 8 Plus. If you are missing data, you can add it manually to the Health app. Open the Health app and find the Flights Climbed category either in your Favorites or in All Health Data. You will see an "Add Data" button in the upper-right corner of your iPhone screen.
Manage Health data on your iPhone, iPod touch, or Apple Watch
"About Flights Climbed
A flight of stairs is counted as approximately 10 feet (3 meters) of elevation gain (approximately 16 steps)."
Source: Health app on iPhone, iOS 13.3
iOS 13.3.1 was released January 28, 2020. Try updating your iPhone and restarting it to see if that makes a difference in the Health app for your future walks.
Update your iPhone, iPad, or iPod touch
The Apple Watch is going to give you a more accurate account of your activity and fitness levels, including monitoring your heart. Take a look at the information below:
Apple Watch Series 5
Apple Watch Series 5 Films - watch the "This Watch Tells Time" video for a quick overview of what the Apple Watch can do.
Take care.