There are many nice places to visit that it can be hard to decide where to go. Here are my 5 recommendation:
Omihachiman old town
Omihachiman has a very beautiful old town part along small canals. You can just walk around the old buildings of do a boot ride on the canals.
If you are already in Omihachiman you should visit the Chomeji temple which is on a mountain north of the City. You can use the bus from the station to go to the mountain. From there you have to climb up 808 very deep steps to the shrine. At the top you can visit the Chomeji temple including a beautiful pagoda. If the weather is good you can enjoy a beautiful view over the lake from a nearby platform.
At the westside of the lake is the city Otsu with it´s beautiful Omi shrine. The shrine is 5 minutes walk away from the JR Otsukyo station. The shrine dates back from the showa period and includes the main shrine and several other buildings. The shrine is located on a hill and surrounded by a small park. If you go there early in the morning you can enjoy a beautiful sunrise from the shrine.
Close to the Omishrine at the Hieizan-Sakamoto Station is the big Hiyoshi Taisha Shrine. You have to walk around 15 minutes from the station through many nice old style buildings and small shrines. The Hiyoshi Taisha Shrine is located at the foot of Mount Hie. The shrine ground contains 7 large and small shrines and 3 small bridges. The main hall was destroyed during a big fire and was rebuild afterwards.
My last recommendation is the Ukimoto or Mangetsuji temple. You have to walk around 20 minutes from Katata Station to the shrine. The Ukimoto or the floating main hall has one of the nicest viewpoints to the lake. The current temple dates back to 1934 and had to suffer many damages from typhoons. Many famous artists and poets visited the shrine and left pictures and poems. Directly at the entrance is a very impressive 600 years old pine tree.