Although annoyingly touristy, Monet’s Gardens are a must. Here are some ways to avoid being a camera-neck-wearing, fanny-pack, matching-t-shirt tourist:
Renting a car: gives you the leisurely luxury of traveling without a strict time schedule.
Walking: After taking the train from St. Lazare skip the bus and bike rentals and travel by foot. The towns of Vernon and Giverny are really charming. You can pop into to their churches, cafés, art galleries, medieval buildings at a whim. Plus there are many great picnic spots along the way!