Wikitravel is always a good place to start:
http://wikitravel.org/en/Georgia_(country)
Click the links under 'Cities' and 'Other Destinations' for loads more information).
Routes to Tbilisi from the UK include the following:
Turkish Airlines from either Heathrow or Gatwick via Istanbul Atatürk airport.
Turkish Airlines from Gatwick via Istanbul Sabiha Gökçen airport.
Aegean Airlines from Heathrow, Manchester, Birmingham (seasonal), Edinburgh (begins 19/06/17) or Gatwick (seasonal) via Athens.
airBaltic from Gatwick via Riga.
Belavia from Gatwick via Minsk.
LOT Polish Airlines from Heathrow via Warsaw.
Lufthansa from Birmingham, Heathrow, Manchester or Glasgow (seasonal) via Munich.
Ukraine International Airlines from Gatwick via Kiev.
Alternatively there are flights from almost every large UK airport to Amsterdam, from where Georgian Airways fly to Tbilisi.