Rent bikes (cash only) at the Café L’Arrivée de Giverny, opposite the train station in Vernon, from where Giverny is a signposted 5km along a direct (and flat) cycle/walking track.
Shuttle buses (single/return €5/10, 20 minutes, four daily Monday to Friday Easter to October, five daily Saturday and Sunday Easter to October) meet most trains from Paris at Vernon. There are limited seats, so arrive early for the return trip from Giverny. Tickets are sold on board; cash or credit card accepted. Check the live shuttle schedule on www.sngo-giverny.fr; there is free wi-fi on board.
There's a free car park near the Maison et Jardins de Claude Monet.
Taxis usually wait outside the train station in Vernon and charge around €15 for the one-way trip to Giverny. There's no taxi rank in Giverny, however, so you'll need to phone for one for the return trip to Vernon. It is preferable to take the shuttle bus back to Vernon.
From Paris’ Gare St-Lazare there are up to 15 daily trains to Vernon (from €9, 45 minutes to one hour), 7km to the west of Giverny. Trains also run to/from Rouen in Normandy (from €7, one to 1½ hours, at least every two hours).