I want to set up multiple alarms in the same day for taking meds on my iPhone 14

You already know how to set up and Save the first alarm.

Once that is done, tap on the + (plus) button in the upper right corner and set up the next alarm and Save

Then tap on the + (plus) button again and set up the next alarm and Save

Etc

On a quick test, I was able to set up 10 alarms, but the + (plus) button still appeared, so 11 is possible and probably much more.

Not sure many alarms that you need.

You could use the Reminders app and create several reminders, at different times of the day, recurring daily.