Paris is ideally placed for day trips. The Paris Basin is large and flat, the rail connections are excellent, and within an hour or two in any direction you can be at a royal palace, a great forest, a cathedral city, or a village that inspired the Impressionists.
Versailles is the obvious choice and justifiably so — the palace, the formal gardens, and the Grand Canal are on a scale that takes a full day to absorb properly. Take the RER C from central Paris (45 minutes, €7.30 return); the palace is a 10-minute walk from Versailles-Rive Gauche station. Book timed-entry tickets online well in advance, especially for summer.
Chantilly (50km north) offers the Château de Chantilly with its remarkable art collection, the extraordinary baroque stables, and a great forest for walking. Fast train from Gare du Nord.
Giverny — Monet's Gardens (80km north-west) is where Monet lived and created his famous water gardens. Train to Vernon from Gare Saint-Lazare, then a short taxi or bike ride. Worth the effort in May and June when the gardens are at their best.
Fontainebleau (65km south-east) has a magnificent royal forest — the largest in France — and a château that was home to French kings from François I to Napoleon III. The forest has exceptional rock climbing and is popular with Parisian weekenders. RER D from Gare de Lyon, then a short bus ride.
Épernay (130km east) is the centre of the Champagne wine region. The Avenue de Champagne is lined with the great Champagne houses — Moët & Chandon, Perrier-Jouët, Pol Roger — most of which offer cellar tours. TGV from Gare de l'Est, just over an hour.