Would it be better to take a car to the dealership or a local garage to service a malfunctioning air-conditioner?

It's a 2002 Honda Civic sedan.

A local garage -- because it's cheaper.

Tbh mate, with that car id get a new one

Depends on the garage or dealer. Plenty of garages are great and plenty of dealers are incompetent crooks.

Local garage

A local garage that advertises a/c work.

Hi best going to a Honda dealer.

You would be a fool to take a 15year old car to a main dealer. They will charge you a 'mint'.
Find an air-con specialist… And NOT one of these cut price 'Re-gas' cowboys.

A garage that does A/C work. Not all of them do, so they would hang a sign saying that they do, do it.

The dealer will always be the most expensive.