Giverny

The metro does not go to Giverny. You take a train from Gare Saint-Lazare to Vernon, a small town a couple of miles from Giverny. Shuttle buses to Monet's house and garden meet each train. You can also rent bikes or walk, don't know about taxis, but the bus is the easiest way.

The train to Vernon, the stop for Giverny is not the metro. You will take the metro to Gare St Lazare and then take a train from there to Vernon. At the Vernon stop, there is a shuttle that takes you to the gardens. I think that is the easiest way from the train station to the gardens. You can also rent bikes and cycle between the station and the gardens. Here is a link to information shuttle: https://giverny.org/transpor/Giverny-bus-shuttle-2017-en-couche.pdf

To the good advice already given, I'll add that I'd recommend you download the SNCF-Connect app which is the official app for the French train company.

Reservations for the train to Vernon to not need to be bought very far in advance, I think I did it the day before I traveled last October. I had the ticket on my phone for both outbound and return. Very easy.

Do look at the time schedule and pick the direct train. When I went there was a direct train at 10:11.

I'd also recommend you download the BOLT app. I usually take the shuttle to Giverny but last Fall the train was late and the shuttles were not all in place so got quite crowded. Friends used the BOLT app (similar to Uber) to go to Giverny and then we returned using Bolt as well.

I became aware during my planning that there are construction works on the train system from Paris to Vernon/giverny on some weekends during the year. One weekend that may affect you is June 14-15. I couldn’t figure out why I couldn’t get a train schedule for the weekend I wanted. So I’m doing a Paris City Vision tour that just takes you there and then picks up later so you are free to do what you want. I’ll report back later if that would be an option

Giverny is not even in the Ile de France i.e. the district that Paris is in and so Paris public transport does not reach it. It is easy to do from Gare St. Lazare. When you book your Giverny tickets you can also book the shuttle between the train station in Vernon and the Monet site. Step lively when you get off the train as the transport does fill up.