It's nice to hear that you have chosen Vilnius when we're celebrating 100th anniversary of Lithuania's state!
First advice - don't take the local taxi from the airport if you don't want to overpay for your trip. You can use an app e-taksi or just find some taxi company and call them (the waiting time will be up to ~5 minutes).
In fact the airport in Vilnius is quite near the center and you can use public transport, ticket can be bought at the bus driver but don't forget to mark it after, price - 1€ (app m.ticket, website vilniustransport.lt):
Buses 1 and 2 can take you to the bus/railway station (both of these are located near each other).
Bus number 88 can take you to 'Aušros vartai' (a first stop from airport) - a nice place to visit and it could probably be your starting point for the journey.
Most of the nice places are located in the old town of Vilnius so you can reach them easily by walk.
Starting from the 'Aušros vartai' you could then visit Town hall, "Vilniaus gynybinės sienos bastėja (The Bastion of Vilnius City Wall)" which is near Užupis. After that you could go to Užupis, find a "Constitution of Užupis republic" written down on the wall, find an angel of Užupis, then reach gothic style Saint Anne's Church, Bernardine Garden, Presidential Palace (there's a pub 'Klaipėdos Senamiestis' near the Presidential Palace where you could find delicious local made cheese sticks with a dipping sauce), Vilnius University, Cathedral, Royal palace, Gediminas castle, Money museum, walk down the Gedimino avenue through the statue of Kudirka (the author of Lithuanian anthem) and the anthem of Lithuania written behind him, walk couple of hundred meters to the right and find the Opera house. Then back to Gedimino avenue to reach Vilnius regional court and find Lithuanian Partisans' names on it which fought for Lithuania's Independence, also Lukiškių square and finally the Parliament of the Republic of Lithuania (Seimas). If you can still feel your legs climb on 'Tauro kalnas' to see a view of Vilnius.
Of course you would also find a lot of different churches, and typical short and narrow old streets on this road.
Local pubs and places to eat - depends on your preferences. There are a lot of places near Gedimino avenue - 'Užupio picerija (pizza)', 'Būsi trečias (beer with flavours of berries)', 'Klaipėdos senamiestis (drinks, cheese sticks)', 'Grill London (grilled meat or vegetables, burgers, drinks. usually they don't provide a reservation of the tables so you might find a queue outside)', 'Talutti (pancakes, desserts)', Cat cafe (just a relaxing place with some food, drinks and cats). You could also find ramen, sushi, indian, mexican, chinese, junk (hesburger, kfc, mcdonalds) food restaurants. If it's not cloudy and no fog you could visit TV Tower (located quite far from old-town) restaurant and see a full view of Vilnius.
That's what comes to my mind at the moment