Anykščiai (population 12 000) lies on the Šventoji river, the longest river to both start and end in Lithuania. Despite not having magnificent buildings (other than the imposing gothic revival Saint Matthew church of 1909 with its tallest-in-Lithuania 79 m twin towers) the town has quite many interesting places for its size. Some of these places are outside the town, however (up to 10 km from the center), therefore a car would be convenient.
What you can see in the town itself besides the church is the narrow-gauge railroad, declared to be a technical monument. Built in the late 19th century this railroad once connected far-away places like Biržai and Švenčionys and had hundreds of kilometers in length, quite unusual for European narrow gauge railways. It was shortened to Panevėžys-Anykščiai-Rubikiai line by the Soviets. However, some of the stations, water towers are dating to the turn of the 20th century. After a brief cessation of service, the line has been partially restored by rail enthusiasts. Tourists may now enjoy a short ride to the nearby lakeside resort of Rubikiai in weekends. A hire of entire train is also possible. A short ride by a handcar or a railbike at the narrow-gauge railroad museum in Anykščiai station is much cheaper and you may also witness old rail machinery and memorabilia there.
Southwest of Anykščiai lies the Puntukas stone, the second largest stone in Lithuania. It is covered by indentations reminding of the flight across the Atlantic Ocean by pilots S. Darius and S. Girėnas, the first Lithuanians to complete this mission (and worldwide air mail pioneers). The bas-relief was made in secret by Bronius Pundzius in 1943 as Lithuania was occupied by the Nazi Germany at the time.
Liudiškės hill, on top of which lies the grave of famous Lithuanian writer Jonas Biliūnas, is nearby. The grave is shaped like the "Phleron of happiness" in one of his short-stories. Everybody sought for that Phleron but it was on a large tower, and many people fell down to their deaths. Only after many died somebody climbed over the dead bodies of the earlier braves and took the Phleron, and made everyone happy (possibly an allegory of a fight for independence, where many have to die before somebody is able to successfully declare it).
North of Anykščiai Šeimyniškėliai hill is a possible location of mysterious Voruta city where Lithuania's first king Mindaugas had been crowned. The massive triangular hill is surrounded by a defensive ravine and may be ascended by wooden stairs. Unfortnuately, no fortifications remain on top. However to help one imagine how the castle may have looked like a wooden tower and outer wall is reconstructed nearby. Tower hosts a mini-museum inside and one may also try out shooting old-style bows.
In Niūronys north of Anykščiai there is a popular Horse museum, dedicated to horses and pre-war countryside lifestyle. Craftsmen (blacksmith, weaver, wheelmaker) show their work and its results. 19th-century horse-drawn vehicles and the cultural role of the horse are also well-covered (mostly in Lithuanian though). The museum is famous for programs where you can try the old village crafts yourself - such as making bread (reservation needed).
Anykščiai placed a bid to become an official resort. It offers active tourism opportunities including a skiing hill. In summer the same hill serves as a place for "summer sled" (a kind of roller coaster with an ability to control its speed) and zorbing. Wakeboarding park and labyrinth park are available elsewhere.