Private clinics in Tbilisi are good, fast and inexpensive by European standards. Many doctors speak English; at the larger clinics you can ask for one when booking.
The emergency number is 112.
Insurance: Georgia's state system does not cover foreigners, so you are buying private cover or paying out of pocket. Paying out of pocket for a consultation is genuinely affordable; a hospital stay is not, and that is what cover is for.
Pharmacies are everywhere and many things sold on prescription elsewhere are available over the counter. Bring a photo of any prescription you rely on — brand names differ.