In addition to some of the suggestions above, a few other places within reasonable distance for day or weekend trips would be:
Some people don't care for Geneva, but I like it; especially its old town. If you have an interest in physics, you can also visit CERN on the France/Switzerland border in a suburb of Geneva.
On the south side of Lake Geneva (Lac Léman) are some pretty towns. Yvoire is one I've visited and plan to return to.
Southwest of Lyon is Le Puy-en-Velay, which is located in the volcanic Massif Central part of France. It's a hilly town with lots of nice views and a cool cathedral that is part of a pilgrim trail (I'm not Catholic, so that stuff doesn't really stick with me). It's known for lentils (lentilles verte du Puy) and lace (dentelle). I recall when I visited Le Puy, at the base of the stairs leading up to the cathedral one of the shops had an old mechanical lace-making machine set up, clacking away and making lace. On top of one of the steep peaks in town is a statue of Mary holding the baby Jesus (Notre Dame de France) -- this was made by melting down a couple of hundred Russian cannons captured during the Crimean War and was given to the town by Napoleon III. If interested, do an internet search for images of Le Puy to get an idea of what it looks like. Here's one that might give you an idea.
North of Lyon and west of Mâcon are the remnants of the vast abbey at Cluny. The whole region around that is peppered with dozens of Cluniac Romanesque churches from the 1000-1200 era; possibly one of the largest is in Paray-le-Monial, which is the site of a large summertime pilgrimage with thousands of visitors, mostly from France but some from outside of France. Paray is about an hour 40 minutes from Lyon by car.
Southeast of Lyon is alpine foothill town of Grenoble, which would make a nice day trip by car (and possibly train). Annecy is another town not too far from Lyon that some people like -- it's situated on the edge of an alpine lake and has some pretty canals coursing through it.
While in Lyon, make sure to visit le marché quai St. Antoine along the Saône River. It's a good and quite large market.