The oldest and largest shopping mall is Akropolis (Šeškinė borough), well known not only to people all over Lithuania but also in Belarus. In weekends it might get heavily overcrowded which leads to lack of parking.
Other large shopping malls are not far away. Ozas, with its extensive food court, suffers from having opened during the financial crisis of 2009 but is more spacious and convenient. Panorama near Žvėrynas is lagging behind Akropolis and Ozas, while mid-sized Europa in Šnipiškės caters to the upscale market.
High streets of Vilnius emptied out to some extent with the advent of the supermarkets but there are still many upmarket shops in Gedimino Avenue (New Town), especially in its Gedimino 9 shopping mall. Unfortunately, parking is expensive there (by Lithuanian standards).
If you want to shop the "old style" there are several bazaar-like open markets. In the suburban Gariūnai next to Vilnius many of today's businessmen started their business in the early 1990s. Just don't forget to negotiate. Smaller market closer to the city center is the very old Kalvarijų market in Šnipiškės. Today there are also a few modern markets mostly aimed at ecological food.
Main supermarkets have working hours until 22:00 or 23:00 and are open 7 days a week. Other shops may close down earlier. Marketplaces are only open in the first half of the day.
For souvenir shopping, there is the ordinary selection of t-shirts, cups, and magnets at the places most popular among tourists: Pilies street and Aušros Vartų street in the Old Town. A viable alternative is to buy your souvenirs at a supermarket - the larger among them have a dedicated shelf.
If you prefer regional souvenirs there is amber (likely to be imported from Kaliningrad but turned into jewelry by local artisans). In Pilies street you can also buy paintings by local artists. Don't expect Michelangelo there, but the prices will be much lower than in the West (if you negotiate well enough).
Arts and crafts are also available there but if you want some real shopping come during one of the fairs (St. Casimir, St. Baltrameaus). Many Old Town streets become large outdoor craft markets at these days.
Given the stories of miracles in Vilnius, you may want to buy yourself religious goods. Christian religious paintings, including replicas of the famous Divine Mercy painting, are sold on the southern side of the Gate of Dawn.