We went to Giverny yesterday and were really glad we went early. It was so crowded!! Probably because most French people take a 4 day weekend to take advantage of the Ascension Day holiday on Thursday (5/25). Seemed like a lot of French tourists there.
So we arrived at the Gare St Lazare station in Paris at about 6:35 am, easily bought tickets at the SNCF boutique in the station from a very helpful agent and got on the 7:19 am train which arrived in Vernon at about 8:05. Also, we got open ended tickets for the return trip since we didn't know which train we'd be taking coming back..
We knew that the Vernon-Giverny shuttle ( 10 euros round trip) didn't start running until 9:30 so we planned to take a taxi from Vernon to Giverny. There were others there waiting so we shared the cab with them (6 total) so it cost 3 euros apiece. You could also use Uber. There wasn't a taxi at the station when we got there so I started to contact a Uber driver, but then the taxi came.
We got to Giverny about 8:30, enough time to wander about, find the bathroom, etc. then we got in line about 9:00 to buy our tickets (we were almost at the front of the line). By the time the ticket office opened at 9:30 the ticket line stretched a long ways down the road!
There is a different entrance for people who already have tickets that is around the corner from the ticket office so you don't have to wait in the ticket line.
When we emerged from the garden there was still (!!) a long line of people waiting to buy tickets!
We took a taxi back and split it with two other people who happened by at the same time so it cost us 5 euros apiece that time which made it 8 euros round trip - cheaper than we would have paid if we'd taken the shuttle. The driver seems to charge 15 euros for 2 people and 20 euros for a group so it's worth it to try to go in with others. Back at the station there was a very long line of people waiting to take the shuttle and about 20 people waiting in the taxi line.
We had about an hour before our train came so we took Rick's advice and walked into Vernon to the Rose Boulangerie for fabulous pastries and delicious sandwiches.
Despite the crowds, we were glad we visited Giverny. We went straight to the pond when we got in and it was just lovely with the beautiful morning light and the birds singing. Then we went in the gardens which had gorgeous flowers. There was a long line to get into the house/museum - we were told it was a 15 minute wait. By then we were done with crowds so we skipped it and left.
So I would definitely recommend taking the train, and specifically the early train, and I would buy the tickets to Giverny ahead of time. Have fun!