I would skip the underground lake at St. Léonard. It is somewhat "underwhelming" and quite a hassle to reach from Lutry! Defintely do the walk from Chexbres down through the Lavaux vineyards to St Saphorin. Very lovely views over to the French alps. And there is a little café in St. Saph that serves good filets de perche - the local lake fish. This is the Café du Raisin. Very very small and you must reserve a table. Unlike many other local restaurants they actually serve perches from the Léman and not from Estonia!!
If you want to see a little of Lausanne (not easy to navigate!) then take a lake boat from Lutry to Ouchy and the Metro to the Place de la Ripponne. Look up and follow your nose up the steps to the cathedral. You will have a great view down over the lake from there and the cathedral itself is a lovely example of Gothic architecture. It has a beautiful rose window and a vey interesting façade, recently restored. It sits in the middle of the historic centre of Lausanne, and there are some lovely 17th and 18th century buildings up there.
Then walk down to the Place de la Palud, where the Hòtel de Ville is located. This is also a lovely building. there is an interesting "clock" at this square and at noon Vaudois gentlement walk along it.
Continuing down you will come to the rue Centrale, the main shopping area. Walk up to the Place st François and then down to the station and take the train back to Lutry.