In Vilnius (also in other big Lithuania cities) it's interesting, comfortable, but not safe to travel by bicycle. First of all there are not lot of cycleways. Riding on the streets it's dangerous, because streets are not in very good condition and drivers are intolerant with riders.
When you ride on Vilnius streets by bike, wear vivid T-Shirts and always use flash light.
Excuse, I didn't go by bicycle around Vilnius but I could see few locals on bicycles. I didn't see any parking lots for bicycles and any rent a bicycle agencies.
I saw only one bicycle trail (on my picture) along Gediminas Prospektas, the most representative shopping and commercial street of Vilnius.
But looking at chaotic at some places traffic I coudn't say that Vilnius was the bicycle friendly city. Maybe soon...
You can rent a bike to cycle around the city, maybe if you actually plan to get a bit further it is a good idea, but just for the old city you do not need it.