You will find the most options for accommodation in the Lithuania’s main cities (Vilnius, Kaunas, Klaipėda). It may be wise to anchor your trip there as the selection is great: from youth hostels with shared bathrooms to five-star chain hotels. Take note that stars are awarded to the hotels based on them having certain amenities and not on the quality of such amenities.
Another good option for staying are the resort towns. There are the most hotels in Palanga, followed by Neringa and Druskininkai. Some of the hotels in the seaside resorts close in winter, but many remain open. In summer they are joined by hordes of local people renting out their apartments or rooms. In Palanga or Šventoji such people line up near entrances to the towns and also at the bus stations.
Resort towns also have a growing array of spa hotels that offer various supposedly healthy procedures, saunas, and pools, and are typically open year-round.
That said, there are hotels in many smaller non-resort towns as well, but there you might be limited to a single option or a choice of two places for spending the night. Motels are available on the major roads, such as Vilnius-Kaunas-Klaipėda or Panevėžys-Šiauliai-Palanga.
Another option is the so-called countryside tourism sites. These ranges from basic farmstead accommodation to elaborate hotels built in the countryside rather than cities or towns. The possibilities there may include horseback riding, a traditional sauna and so on (but check this in advance). It is popular among Lithuanians to hire such farmsteads for celebrating various events.
B&B (Bend and Breakfast) is a term that even lacks Lithuanian translation, although you may find similar deals going by other names (usually "hotel", "villa" or something like that). Hostels are limited to main cities.
Campgrounds are scarce by European standards so it is better to check in advance the locality you plan to stop at if you need one.
The accommodation in Lithuania is less expensive than in the northern Europe but more expensive than in many eastern countries. VAT adds to this.
Hospitality Club, Couch Surfing and similar institutions of exchanging free stays with locals are popular among Lithuanians.
Except for the resorts apartment rental is generally useful only for longer periods (i.e. months). Downtown rental tends to be 25%-40% more expensive than the Soviet districts rent and Vilnius rent is 25%-40% more expensive than Kaunas or Klaipėda rent. Free apartments may be hard to come by in the largest cities in August-October (when the university students move in). Lithuanian law is generally favorable to the tenants (e.g. allowing to cancel the deal easily), but you may find it hard to defend your rights if you won't sign a written contract.