Gudauri is located about 120 km from Tbilisi and it usually takes around 2 hours to get there from the capital. The cheapest option is to take a local minibus (marshrutka) from Didube bus station. Take the red line metro in the city centre and get off at Didube. The bus station is big and can seem chaotic. Expect to be approached by drivers or their touts, so if you don’t speak either Georgian or Russian it could be daunting trying to find the right bus. In reality, ask around for “marshrutka do Gudauri” and you’ll quickly be directed to the right area. There may be several heading that way, there is no fixed schedule, they leave as soon as they are full. They depart from 08:00 to 17:00 and the ride costs 7 Lari. In peak-season the marshrutkas can fill up fast, so give yourself plenty of time for the journey.
Some buses go further along the route to Kazbegi, so you might need to remind the driver to stop at Gudauri - the hotels and chair lifts will make it pretty obvious when you are in the resort.
An easier, though more expensive, option is a taxi. Expect drivers to ask for around 100 Lari for a one way trip, but try to negotiate and you could get it cheaper.
There are also several companies providing transfer services, some of which offer services direct to Gudauri from Tbilisi and Kutaisi airports, or ask the hotel you are staying at in Gudauri if they can to organize transfers for you.