"On and Off the Road in Monet's Garden in... " a Giverny Travel Page by Roadquill

Trains leave the Gare St. Lazare a number of times during the day and stop at Vernon, the jump off point to Giverny. They have transportion information at the information booth at the train station. They give the times the trains depart from Paris and when they arrive at Vernon and coordinates when the bus leaves Vernon and drops you off in Giverny.
Round trip train cost was 22 euro for U60 and 18 euro for my parents.

Once in Vernon, there is a mad dash to the bus (to the right of the staion as you leave the station) as many times there are more passengers than the first bus can take. Bus cost is around 3 euro RT. Taxi (in front of the station slightly to the right) one way is 11 euro. My parents, in their 70's, took, a taxi there and the bus back. My wife and I rented bikes (t the right and across the street from the station) for 10 euro each for a day's use.

Since bus and taxi are pretty routine, here is the bicycle tips.

There are several places that rent bicycles. The brasserie furthest away seemed to have the best price and were very pleased that we walked an additional few yards for their services. They gave us a rough map, a basket and sent us on our way. They don't really speak English but they point well. The path follows the bus starting point (where those unfortunate enough to miss the first bus still waited), down 30 years and then you make a left an ride through Vernon and go over the Seine River. You can follow the road to the right towards Giverny or you can continue on another short ride and there is a bike path.

This is where we rented our bikes, and then it was the perfect place to have a cold brew afterwards.