The definition of a "day trip" from Paris often differs depending on who you talk to.  One can definitely go as far as Le Mont St. Michel in a day, and even explore the Chateau of the Loire.  I, however, prefer to refer to those as "side trips", as it is my personal opinion that anything more than 50km outside of Paris is best left for those who have time to overnight outside of the city, and enjoy those areas a little more in-depth.  The Day Trips listed below are all designed to allow you to leave your hotel or apartment around 8am, and return to the city in time for dinner.

St. Germaine-en-Laye
Chateau, gardens and museum.

St. Germaine-en-Laye is the birthplace of Louis XIV, the Sun King, and an exquisite example of 12th and 15th Century architecture.  The chateau now houses the Musée des Antiquités Nationales, and although you won't find the rooms decorated as they were in the days of French royalty, you will still get an idea of the chateau as a residence.  The entry fee for the museum is currently €4, and they often have temporary exhibits that you will be able to see with your museum entry fee.  Louis XIV lived in this chateau for much of his young life, having fled from the Louvre during the Fronde.  It bore both fond and bittersweet memories for him.  In later years, he swore off visiting the place because from the balcony of his corner bedroom, he could see the towers of the Cathédral (aka Basilique) de St. Denis, where he knew he would someday be laid to rest.  The vision was too painful he said, and so he never returned.  Here, you too can see the extraordinary view the Kings once cherished.  Although the Parisian skyline is vastly different, and today's smog ridden air makes a clear view difficult, you can still make out some of the cities more prominent landmarks from the edge of the gardens, including the Eiffel Tower and the Sacré Coeur.  Alas, St. Denis is no longer in view, but you'll still be able to get a sense for what the Sun King felt.
After spending your morning in the museum, we recommend lunch.  If the weather is warm and dry, you may choose to purchase a sandwich from the vendor on the grounds.  Enjoying your lunch in the outdoors in this most exquisite garden is a real treat.  You may, however, choose to dine at the Pavillon Henri IV, 21, rue Thiers, on the chateau grounds.  Their menus begin at €60 and go up to €90.  It's not cheap, but the view is unparalleled.  After lunch, prepare to spend a few hours in the chateau's gardens.  Designed by Andre le Notre, one of Frances most famed garden designers (Versailles, Tuilleries, Vaux le Vicompte), it is a stunning example of his divine gift.

Getting there.  RER A goes directly to the city of St. Germaine-en-Laye, and can be accessed from the Paris Metro.  Buy a round trip ticket.  When you exit the RER platform at St. Germain-en-Laye, use the exit marked for the Chateau, and you will find yourself directly in front of the gardens, with just a short walk to the entrance.  If you goof and take the wrong exit, don't worry, you'll just be across the street.   

Note:  The museum is closed on Tuesdays.

Once your tour of the garden is complete, you may choose to spend a little time in the town of St. Germain-en-Laye, and return to Paris at your leisure. 

Versailles
Chateau and Gardens
Official Website
Official Website (Limited) In English

If you haven't done a lot of reading about the palace, and especially if it's your first time, I would highly recommend a guided tour. First of all, however, please be aware that Versailles is uber-strict about who can conduct tours in the chateau. Only their own guides, and accredited (by the state) guides outside the chateau can lead groups. The list of others is quite short. There are a number of companies who offer private tours of Versailles, but it's important to know that their roll in your "private" tour stops at the front door, when they hand you over to a Versailles employee. Be certain if you book anything that you're not just getting a ride to the palace and back. On the flip side to that, companies like Cityrama will load up a bus in Paris with an accredited guide that will take you through the chateau as part of their tour, and they often dispense a wealth of information you'll never get doing it on your own.

With that said, there are four public entrances to the chateau. All are pretty clearly marked with the letter A, B, C & D. Each entrance offers a different type of tour. Entrance A is for those people purchasing tickets for the State Apartments, B is for Tour Groups, C is for Louis XIV's chambers, and D is for various guided visits, including the Opera House & Chapel, Marie Antoinettes private apartments, etc. Audioguides are available for all of the public areas. There are pamplets located around the entrances which will help you decide the type of tour (self guided or escorted) you want to take. Some parts of the chateau can only been seen on a guided tour, and tickets must be purchased in advance (in advance in this case means before the tour starts...the earlier the better, as they fill up quickly).

Most people who want to spend a full day at the chateau will benefit from purchasing the One Day Pass for 20 euros. The pass (which can be purchased at the Versailles ticket office, FNAC stores, or at some SNCF stations) will grant you entrance to just about everywhere you want to go, including the gardens. It does not cover guided tours to Marie Antoinette's apartments, or the Opera House (both must be done on separate guided tours). It does get you into the wing where the chapel is, and you can peek in, although you can't go in (going in is on the Opera House tour). The one day pass can also be combined with your train ride out to the chateaux with a special purchase price if you go to an SNCF ticket office.

The Fountains
Typically, they play every Saturday and Sunday from the end of March until the end of September, from 10 until 12, and again from 3 to 5pm.

Whichever way you decide to go, have a fabulous time. And remember, as you walk though the palace, take a moment to turn your attention away from the gold leaf, the Rigaud paintings and the ornate architecture, and look down at the parquet floors. The chances are pretty good that someone very important once stood exactly where you'll be standing.

Vaux le Vicompte and Fountainebleau
Chateaux and Gardens.  Car rental required.