Georgia has been more culturally and religiously linked with Russia than other Caucasian countries. The great majority of the people speak Georgian and are nominally Christians of the Georgian Orthodox Church. Shaking hands with both sexes is fine, though it's polite for a man to wait for a woman to extend her hand. Say Gumarjoba (‘Hello') as you do so. When a relationship becomes closer, you may expect a kiss on the cheek. On initial meetings you should call people by their surname with the title Batono (Sir) or Kalbatano (Madam). From then on first names can be used prefaced with the appropriate title.
This is still a very hierarchical society, where both important and older people are treated with considerable respect.
Hospitality is central to Georgians. Guests are to be treated with respect and generosity. As a visitor from another country, you can expect a warm welcome and it's a matter of honour that you will be paid for in a restaurant. Never offer money, which may offend. A present, especially from home, is a better idea.
If invited to a Georgian home, be sure to accept. It's polite to bring a small gift: pastries, sweets (especially imported ones) or flowers are good, though flowers should always be in an odd number. Keep your elbows off the table and clean your plate. You are likely to be offered more, at least once.
Tips are much anticipated and appreciated, usually in the region of 10-15% of the bill total unless service is already included.