What are my options for travelling from Western European cities to Tbilisi, Georgia by air?
The Wikipedia page for the Tbilisi airport lists the airlines with flights to Tbilisi. Looks like your best Western Europe options are Rome, Amsterdam, Paris (seasonal) and Munich. There are other flights from Eastern Europe.
If you're looking to get in cheap, Wizz Air (a discount airline) serves Kutaisi from various locations in Eastern Europe.