Is Chillon the castle just up from Montreux? If so, you also have Vevey nearby, on the same train line as Montreux, where Charlie Chaplin spent 25 years of his life. It's been a few years since I've been there, but I do recall a nice outdoor market on Saturday, and a nice walkway along the lakefront. I was fortunate to do it by lake steamer, but those would not be running this time of year. You'd certainly be able to pop around between towns in that area by rail.

I checked the Swiss rail timetable and it looks like both Zermatt and Chamonix are about 3- 3.5 hours from Geneva. If you're going to head for one of those, I'd pack a lunch and bring it along on the train. When I was in Chamonix (on a French rail pass) the weather was so bad (blizzard conditions) we could not go up in the tram off the valley floor. Zermatt, as the entire ride is by train, would not have that issue.

The last hour of the train ride, from Brig to Zermatt, is quite scenic, and Zermatt itself is a vehicle-free town, and you can hike up a trail at the top of the city in the direction of the Matterhorn. I did this about 3-4 years ago and stopped at a cafe along the trail for lunch, along with skiers coming by (a very small cafe). If your view is not blocked by clouds, the Matterhorn would be very visible.

You might take a look at the Swiss Rail timetable and see what options you have:
Swiss Rail

If you're looking to visit another city, Bern (are the bears still there?) and Lausanne are both good recommendations.

Sadly, you're too far away (5.5 hours) for the Jungfrau or Kleine Scheidegg, over in Rudi's territory.