There are really four ways to get to the airport: by public transport (bus #1, #3G), shuttle bus from the Bus Station, express railway coach from the Railway Station, and by taxi. The distance between the airport and city centre is short (~7 km), so in theory you should get there pretty quickly (<30 min), but during rush hours and on bad days your route can be clogged by traffic jams, then it can take up to 60-80 min to reach the airport.
My personal favourite is taking a train to the airport. It only takes 7 min, costs ~$1, and the most reliable means of transportation. Plus you can comfortably walk to the Railway Station from the Old Town. You can plan your trip here. As you can see, the downside is that these trains run only once an hour.
The second best way to get to the airport is to catch a shuttle bus from the Bus Station, which is located just across the street from the Railway Station. The buses run every 40 min and take around 10-15 min to get (once again subject to the traffic conditions) and cost around $1.15. Here is the schedule. Beware that number of seats on the shuttle is limited to 15-17.
Public transport option is a bit impractical, in my eyes, since you would have to buy an electronic ticket holder card, then top it up with credits/tickets and in the process figure out the Byzantine payment/usage system. Needless to say, it will be slower, more expensive, and less reliable route than any of the above. Here is the link to public transport website just in case: http://m.stops.lt/vilnius/#vilnius/en
Getting to the airport with a taxi used to be a hit-and-miss affair, where the cost of your ride was directly related to the callousness of your driver, and the price of taking the taxi from the airport would always exceed the price of the taxi to the airport by the factor of 4. Luckily, the airport management seems to have finally broken the racket that was going on and now it is finally possible to get the taxi for the same price to and from the airport.
As a Lithuanian living in Vilnius centre, I hardly ever pay more than 25 Litas (~10 USD) for this trip, so if you pay over 50 Litas you are being ripped off. The most reliable and hassle-free taxi is "Vilnius veza" - they have a fleet of new cars, present you with a receipt and accept credit cards. You can haul a cab without incurring any extra costs by calling 1450 (on a local network) or +370 5 233 3337. For inquiries/complaints write to firstname.lastname@example.org Alternatively, call +370 5 266 6662 - their cars are older & they don't speak English, but they give you receipts on demand and generally are OK.
Finally, to those of you who think that going to the airport by foot can be fun. I have bad news for you - the distance itself might be relatively short, but besides the fact that the route goes through some of the ugliest & dullest parts of Vilnius, the adjacent areas are known for high concentration of junkies, loitering hoodlum, and garden variety drunks who are calling Naujininkai and Kirtimai their home. People speaking English, looking foreign, or holding shiny gadgets are met with lively curiosity and a graphic request to lend local gents your personal belongings for an indefinite amount of time. Emergency phone number is 112 :)