Take a look at the most interesting of them and go beyond the touristic area.
Don't go to the Palace of the Grand Dukes. Just don't. It's an embodiment of corruption in Lithuania, so don't support it with your money :)
If you speak Russian, it will help you, but don't assume that everybody speaks Russian. Most people in Vilnius do, but you should always ask about it. When you ask them and they say they do, smile to show that you appreciate it - rather than think they are demented if they can't speak good Russian after their country was in the USSR for years. If you speak Polish, that also may help, but again, don't assume that everyone in Vilnius must be fluent in Polish (and most people are not). Polish tourists who approach people and speak Polish annoy many inhabitants of Vilnius. So the best strategy would be to ask them if they speak any of the languages you know and let them choose. Remember that Vilnius is a student city, so many people come from places where they have never encountered any foreign language, aside from those they learnt at school (English and German or French). Whichever language you speak to them, show your appreciation, express that you think they are smart multilingual people, not colonial subjects who should automatically speak the languages of their big neighbours.
Don't leave your money in an easily accessible place in your bag. Pickpocketing is quite common. Never leave your bag open on a crowded bus.
Try not to let taxi drivers rip you off. Airport-centre ride should cost no more than LTL 20 in the current prices. I used to go to the airport at night from Lazdynai, a residential area, for LTL 14 (EUR 4).
Don't assume that the food they serve in such restaurants as Cili kaimas or Fortas is what Lithuanians eat. Most families would eat this heavy potatoe stuff only on holidays, as it takes ages to cook (if properly cooked, though, this food tastes good, and I know many foreigners who absolutely loved it). You will see that what they market as traditional Lithuanian food is like Eastern European Jewish food de-kosherised (i.e. pork and sour cream added). I recommend mushroom (boletus) soup, cold borshch (although its pink colour scares some people), and pancakes (all my foreign friends liked my pancakes with apple, but I don't know any restaurants that serve something of that sort). In Fortas, mushroom soup in a loaf of bread is my absolute favourite. Balandeliai (minced meat mixed with rice, wrapped in cabbage leaf and boiled) is what I also find good.
Avoid saying 'in this country' or 'in Eastern Europe' with a tint of patronising in your voice (like, 'So I know corruption levels are very high in this country, right?'). We all know what is wrong. But try to leave your prejudice, otherwise people get defensive and you won't hear the interesting stories they have to share about the wrongs you know.Forget everything you've heard about Eastern European women. You might meet quite some of those who match the stereotype, but you will get on people's nerves if you generalise about it. The way some tourists behave annoys many women, including myself. Many of us are smart and independent enough to be as distant from your stereotype as Scandinavian women (almost :)). We don't jump into bed with anyone who buys us a drink, and Italian accent is not enough for that purpose either.
Don't buy kebabs in Lithuania if you come from any country where people know how to make them. Even if you go to this Turkish guy's (quite trendy) kebab shop at Traku str., beware: they put pork in it.
Don't expect people to smile at you if you don't talk to them. I know this scared one American friend of mine, but that's how it is - most of the time Lithuanians look very concerned about something.
I'll add more when I have more ideas.Do go to Uzupis and climb the hill behind a secondary school you will see on the left as you go deeper into the neighbourhood up the main street. You will enjoy it. In general, take your camera to Uzupis and make sure you have enough space on your memory card. Go along the river to see the gallery with wall paintings and some interesting statues in the river Vilnele. Then go as deep into Uzupis as you can to see amazing wooden houses.
Spend some time at Coffee Inn (esp. in Vilniaus str. or Gedimino str.) just watching the people who come there. You will see many interesting and quite generalisable types. You also have a chance of meeting some young bloggers.
Beware: there are hardly any public toilets in Vilnius, let alone signs showing where they are. Use shopping malls for that purpose. WC is free and opens around 10 am. Alternatively, plan your meals and coffee breaks in order to use it in the places you eat in.
Spend some time at the Cathedral Sqare around sunset. You will see the main subcultures.
If you like salad, check out 'Mano guru' on Vilniaus str. It's a pleasant cafe with a social agenda.
Don't miss one of the marketplaces. I like the Karoliniskiu market. Many people like Hales market, which is in the centre. The legendary Kalvariju market is huge, but it's a bit too much for me, and pickpocketing is so common that one friend of mine jokes that goods circulate very rapidly: the next moment you see some of the things you've bought is missing, it's being sold again.
Learn a few words in Lithuanian. Generally Lithuanians are very surprised to see a foreigner knowing anything about them, so why not make a pleasant surprise? Also, you might want to read about the successes in basketball.
Have an MP3 player with you if you use public transport before 9 in the morning or after 3 pm. It will calm your nerves in traffic jams.
In summer, go to the White bridge (Baltasis tiltas) to see the forms of leisure people enjoy there.
If you are lost or need to ask something, look for students. Most likely they will know English and be happy to practice it. They may also have tips on where to go and what to do. Many of them did not grow up in Vilnius, so they were exploring the city almost like you do quite recently.
Vilnius is strong in classical music - check out the events.
Google what kinds of museums are there. Many foreigners enjoy the KGB museum. Apparently there are museums of police, post, accounting, etc. (thank you Vytautas) Can't wait to check them out when I return to Vilnius. Also there is an amber gallery near St. Michael church - entrance is free (thank you Domas).
Don't miss the St. Peter and Paul church, its interior is really something.
Try to climb the hill of Three Crosses.
On a summer weekend, take a walk from the Cathedral past the Bernardine church towards the Uzupis bridge and observe Lithuanian wedding 'traditions'.
Take some ice-cream and sit at the bottom of the Adam Mickiewicz monument (if you don't understand neither Lithuanian nor Polish, take a local interpreter with you) and amuse yourself hearing the totally different stories that Lithuanian and Polish guides tell about the contested poet.
In winter, have a dessert at Pilies kepyklele - the desserts they serve are good, and it's a good spot to observe people.
In winter you could also take a ride to one of the residential areas and observe how people enjoy snow (or not).
See the statue of the Gaon of Vilnius (Talmud scholar) who looks like Karl Marx, and the statue of Vincas Kudirka (the author of the Lithuanian anthem) who looks like Lenin.
Go to the canteen on Jogailos str. that says 'Sultiniai' for super-cheap local food.
For going out, you can choose Havana Social Club, it's cool.