Vilnius — 130km south, about 1.5 hours by bus. Lithuania's capital is the obvious major destination from Panevėžys: the baroque Old Town, Užupis, the KGB Museum, and a far wider range of eating and drinking options than you will find in Panevėžys itself.
Riga — 200km north, about 2.5 hours by bus. Latvia's capital is one of the great cities of the Baltic, with an extraordinary Art Nouveau architecture district, a large medieval old town, and a strong food and culture scene. An easy day trip or overnight from Panevėžys.
Anykščiai — about 50km east. A small town known for the Horse Museum (one of Europe's more unusual specialist museums) and a famous narrow-gauge heritage railway that runs through forest. Worth combining into a loop rather than a straight return trip.
Biržai — about 55km north. A small town with a Renaissance castle, a distinctive landscape of karst sinkholes and lakes, and a long local tradition of beer brewing — the Biržai region is known as Lithuania's beer country, and the local brewery has been operating for well over a century.