One of the nicest parts of Lucerne is the tree-lined lakeside promenade. It stretches from the Luzern train station to a large grassy park, about 40 mins walking.
Since the path parallels the bus line, you can easily ride back to the Luzern old town if your feet get tired. At the very end, you can catch a train or bus at the Luzern, Verkehrshaus stop to ride back to the Luzern train station or old town.
The path starts in front of the Lucerne train station, crosses the main car bridge, then hugs the lakeshore for a long while. You'll have wonderful views across the lake to Mt Pilatus. There are a couple parks along the way, a playground, and many benches for lounging.
There are also food stands and cafes sprinkled along the path, including Seebad Luzern, a fun swimming spot with a cafe on the roof.
You can walk as far as the Lido park, a big grassy area along the lake. Next to that is the Strandbad Lido Luzern, a family-friendly swimming area with a shallow swimming area, playground, cafe, changing rooms, and other services (entrance fee).
The Swiss Transport Museum is also across from the park.