So, in the USA, one might:
call a car service on the telephone and arrange for them to arrive at your home.
step outside and hail a cab from the sidewalk with a raised arm.
walk to a taxi stand and get the first cab in line, or wait until you are first in line!
Those are the only ways I know of that a Muggle could get a cab in the USA in 1998. And not all of those ways would be available in every place. Nowadays, of course, one could add Uber and Lyft, but those weren't available then. ;)
So, how would a Muggle catch a cab in London in 1998? Hermione will explain it all to Harry, so he will know just what to do. (When you tell me, that is. ;)
ETA: Oh! I need to know whether or not a cab driver would ever be called a "cabbie." If not, was there any other informal title I could use? Or is it just "taxi driver" and nothing else? :)