Based on numerous trips throughout my life, e.g. a several day trip to Paris in January 2006.
France is probably the country that I know best in Europe. I spent most of my vacations there as a kid and continued to travel to France during my high school and university years.
The Bretagne area in the north of the French Atlantic coast is very pretty with a scenic coastline including both rocky and sandy parts. The tides are very impressive and there is lots of seacombing, exploring tidal pools, walking to islands off shore during low tide etc. Be sure to visit Mont St Michel, a picturesque village on a hill that is surrounded by water during high tide. Bretagne is a very green part of France and the weather is somewhat similar to the British Islands.
Aquitaine (roughly the area between Bordeaux and Biarritz) is characterized by endless sandy beaches, high waves (impressive surfing), sand dunes and large areas of nice forest behind the dunes (mostly pinetrees, ferns and oaks). This was my summer vacation territory for many years as a child. In the summer, the weather is great for spending all your time outdoors. The beaches vast and relatively uncrowded (particularly in September), and you can either camp or rent a nice bungalow in the forest behind the dunes. An absolute must in the area is La Dune du Pilat (a giant sand dune that makes you feel like being in the desert, with amazing views of the blue ocean and the green forest. Consider staying overnight, so you can enjoy the sunset, night sky and sunrise from the dune (there are several campgrounds right next to it). Note that the beach right at the dune is not ideal for swimming (dangerous currents, no life guards). Nearby Cap Feret makes for a nice daytrip (though it tends to get very windy on the beaches there. Seignosse Le Penon (a couple of hours further south) is a great beach resort with nice bungalows (there are also some more remote campgrounds in the area). The coast of Aquitaine is also nice for cycling (we once spent to weeks cycling from Bordeaux to Biarritz, stopping many times along the coast and hanging out on all the different beaches which was great fun).
A particularly nice beach resort is Argeles Plage, close to the Spanish border. Nice long sandy beach with a great view of the Pyrennees mountains. The nearby Coullioure is a very picturesque fishing village and should not be missed. Inland from Argeles is the very pretty town of Perpignan. Further east along the coast is Montpellier which is a very nice historical town at the foot of the beautiful Cevennes mountains.
Halfway between the Altantic Ocean and the Mediterranean Sea, the Pyrennees mountains are absolutely great hiking territory. There are numerous several day hikes that can be easily combined. The GR20 (a long distance trail) also runs through the Pyrennees. There are many huts (conveniently located a dayhike or less apart) where you can eat and sleep (they are called gites d'etapes). The food in the gites is very good, and there are often different levels of accomodation (ranging from bunkbeds to private rooms). We once did a 7 day hike in the Pyrennees and greatly enjoyed the amazing scenery during the day and the great French cuisine at night.
The Cevennes in the south of France are also very nice. Note that it can get very hot there in the summer (the whole area is very dry and trees are sparse), so best to go in the spring or fall.
At the border to Switzerland, the Jura mountains are very empty and scenic. And the French Alps are of course absolutely amazing, particularly the area around Mont Blanc. Both the Jura and the Mont Blanc are covered in more detail under Switzerland (since I mostly explored them from Geneva as a basecamp).
Needless to say, Paris is one of the greatest cities in France. However, there are numerous other beautiful cities in France, e.g. Aix-en-Provence, Montpellier, Perpignan, Bordeaux and Biarritz.
Information: Get the public transport map (available at any metro station), pick up some stuff from the tourist office (e.g. Paris est a vous), get a great map (e.g. Paris Pratique par quartier, a booklet of maps by district available from "tabacs"), and pariscope (weekly entertainment guide) available at tabacs.
Accomodation: For a cheap but very central one star hotel, try Hotel Le Central at 6 rue Descartes in the Quartier Latin (Tel. +33 1 46 33 57 93).
Transport: Get a Carte Orange which is valid for one week (bring a passport sized photo with you).
Restaurants: Chez Jaafar at 22 rue du Sommerard in the Quartier Latin is a great Couscous restaurant (you may want to consider making a reservation as they only have a few tables, Tel. +33 1 46 33 95 40)
Viewpoints: Go up the Eiffel tower (check so see if you can walk up - much cheaper), but go during the day, not at night (unlike New York etc., the view of Paris at night is not so impressive as not many buildings are lit and you are quite far away from many of the main sights, making it difficult to identify them at night). Another nice view is from Centre Pompidou (to avoid queues and entrance fees, take the elevator up to the fancy restaurant at the top - just tell the elevator guards you want to make reservations for the restaurant...). The view from the Samaritaine department store building (Metro: Pont Neuf) is also supposed to be great (but it was closed for renovation in 1/2006). The towers of Notre Dame are another good though expensive option.
Along the Seine: Take some time to walk along the Seine. In particular, explore Ile de la Cite (Square du Vert Galant, Place Dauphine, Square Jean XXIII) and Ile Saint Louis.
Recommended quartiers: Apart from the Seine islands, Quartier Latin is great (roughly the area around the Sorbonne), and so are Saint Germain des Pres and Le Marais (roughly the area between Place des Vosges and Centre Pompidou).
International Organisations: For those into IOs, UNESCO is at 7 Place de Fontenoy (7 arrondissement) and OECD at 2 Reu Andre Pascal (16 arrondissement).
One of my favourite places in Europe and also one of the last remote "wilderness" areas left in Europe. Extremely varied with beautiful coastal and mountain scenery which make the island a hiker's paradise. There are also many scenic train rides across the island. If you would like to spend some time on the beach, go to Calvi, which is beautiful little historical town with a long sandy beach on one side as well as a red sandstone coastline on the other side of town. From Calvi as a basecamp, you can do some great day trips, e.g. by boat along the coast. This part of the coast is actually a protected marine park where you can watch Europe's last sea eagles and even dolphins if you are very lucky. The boat trip stops for lunch at a remote tiny village with no road access. The village can also be reached by foot (it's actually an overnight stop on the long distance trail "Entre Mer et Montagne", also called "Mare y Monti" in Corse). If you just want to do a section of the trail, you can do one way by boat and one way by foot and spend the night at one of the two gite d'etapes in the village. However, I would highly recommend doing the whole "Mare y Monti" which takes about week is probably the best hike in Corsica with parts along both the coast and in the mountains (often on the same day...there are amazing vistas, with the ocean on one side and the snow-capped mountains on the other side. Apart from the "Mare y Monti", there are also three long distance trails connecting the eastern and the western coast of Corsica called "Mare a Mare" (we once did the Mare a Mare Central which was nice, passing many remote villages in the hinterland. And then of course the is the famous GR20 for more hardcore alpine hiking (with basic shelters rather than comfortable gite d'etapes).