Here are my top 3 things to do in Vilnius:
Visit Užupis, a cool bohemian part of town where you find both nice bars and restaurants. The district declared itself an independent republic and could be compared to the Freetown Christiania in Copenhagen. The neighborhood is popular for artists and you'll find art galleries and street art almost everywhere. Go to the local bar Snekutis and have a local beer - or their local mead.
Try the local cuisine. It might not seem like the most delicious food, but will offer you an interesting cultural experience. Especially the traditional "cepelinai" is a must (potato dumplings filled with meat). If you are a cheese fan like me, try the local cheese boards and wine. The Lithuanian cheeses might not be comparable to the French or Spanish ones, but I still found them interesting, especially the smoked cheeses. Besides, you only pay a few euros for a big plate of cheese.... Something else that you should try is the cottage cheese pancakes, and the potato pancakes!
Visit the Three Crosses. From this monument you have a great view over the old town (or actually the whole city). It's totally worth the hike up to the top (or if you are lucky, you might get a car ride up the hill. Sam's old classmate who is from Vilnius was kind enough to bring us on a small sightseeing tour with his car).