There is no subway or light rail in Vilnius (although there are talks on the construction of rapid transit).
Vilnius has an extensive network of public buses. They reach even the most remote areas of the city as well as low-rise suburbs. The timetables to some areas may be scarce, but they are never rarer than once in two hours and usually at least one bus an hour. On the most popular routes, there is one bus every 10 or 15 minutes.
Trolleybuses generally travel on most busy routes and their timetables are more frequent. During the morning and evening rush hours there may be a trolleybus every couple minutes on certain routes.
Even if you don‘t know schedules it is fair to expect a trolleybus to come to a stop in next 10 minutes at the latest. This is not so with buses as many bus routes are thinly served. Therefore if you have no interest in checking schedules in advance choose trolleybuses.
In 2013 fast buses were introduced, their routes marked with letter "G". They are as frequent as trolleybuses and somewhat faster than regular public transport as they stop only at some half of all stops en-route. Note that the same numbers are reused for the bus, trolleybus, and fast bus routes (i.e. a bus no. 1, a trolleybus no. 1 and a fast bus no. 1G are not related at all).
Same one-time ticket (30 min or 1 hour with transfers possible) applies for all public transport. You can buy them at kiosks or from the driver. If you buy it from the driver it costs approximately 25% more (this money goes to the driver as a compensation for the additional job).
As for monthly tickets, there are both ones that apply to both types of transportation and ones that apply either to buses or trolleybuses alone. There are tickets valid for several days useful if you will use public transport extensively.
The final public buses and trolleybuses leave at around 23:00 and in some stops it can be as late as 23:40. There are no night buses but the downtown-airport bus operates between some 5:30AM and 2AM.
The timetables of Vilnius public buses, trolleybuses and some private buses are available here. A powerful journey planning tool is also included in this multilingual website.
During the rush hours (~7:00-8:30, 17:00-18:30) avoid driving towards downtown as main thoroughfares get clogged.
Traveling by taxi is not recommended as taxi drivers are known to cheat people, especially (but not only) foreigners. They inflate prices as much as 10 times.
Vilnius downtown has an automated bicycle rent system where a short rent is free. Look for orange bicycle racks.