Minibuses are locally called marshrutkas, and they operate on established routes. After finding out the number of your route, flag down a marshrutka on the street by holding out your hand, palm facing down.
There are also minibus lines from city to city. Their routes end usually at bus stations and city markets. Their destination is written in Georgian, on a sign in the front window. Ask marshrutka drivers if you can't find the minibus you are looking for.
Taxis in Georgia are the most convenient method of travel, and they are very cheap. Trips within Tbilisi range from 3 to 5 lari, depending on distance, and you can negotiate a price with the cab drivers. The vast majority of taxis in Georgia used to be unofficial "gypsy cabs," driven by anyone looking to make some money. Such unmarked taxi service in Georgia was safe and widely used by foreigners living and visiting the country. Drivers would, however, exaggerate the price for foreigners -- it was best to establish your destination and price before getting in the cab. Situation changed a few years ago when all official taxis were obligated to install meters with fixed rates.
To get to the more remote regions of Georgia e.g., Dusheti, Khevsureti, etc. without a tour company, buses and taxis will only take you so far. At some point it will become necessary to hike, catch a ride on a goods-transporting truck, or hire a jeep. Catching a lorry requires that you are flexible in your travel plans. Hiring a jeep can actually be quite expensive because of the high cost of gas caused by scarcity in the remote regions. To find out about either option, ask around at the bus station or central market of the last town on the bus or marshrutka line.