French is the official language, and all the locals will speak it. Some may use some non-standard expression, but most will make the effort not to use them if you are foreign.
Local expressions you might encounter are 'Tantôt' (Soon), meaning either this morning, this afternoon, tomorrow morning/afternoon or yesterday morning/afternoon, depending of the speaker, so ask for details.
As Normandy is a premium tourist destinations, many of the younger people will speak English, and will be willing to speak it. Spanish, Italian, and German are also quite widely studied at school.
Although there are Norman languages, they are mostly dying out, and the speakers will also speak French. You may also meet the occasional speaker of the neighbouring regions' local languages, such as Breton or Picard, but in any case, a stranger would address you only in French (or maybe English if you were in a tourist area).
Rail is the most commonly used public transport in France for inter-regional travel. It is cheap, fast and reliable. Check out reductions for under-26, over-25 and group travellers. Tickets can usually be bought abroad, on the internet, at stations; in advance or on the day.
Caen is the main station, alongside Lisieux, Bayeux, Trouville-Deauville and Cabourg-Dives. There are also stations in Lison, Le Molay Littry, Audrieu, Bretteville Norrey, Frénouville Cagny, Mézidon, Moult Argences, St Pierre sur Dives, Coulibœuf, Le Grand Jardin, Pont L'Évêque, Blonville Bennerville, Villers/Mer, Houlgate and Dives Port-Guillaume.
Trains go towards Saint-Lô (Cherbourg and Rennes), Paris (2 hours away), Alençon (Le Mans), and Rouen. SNCF website
Roads in France are good.
The main motorway is the A13 to Caen from Paris (225km / 139 miles). It then continues to Cherbourg (although it is not always a motorway). Some of it is toll, but quite cheap. The A84 goes from Caen to Rennes. You can also take the RN13 from Paris, which is free.
To cross the Seine, you can use the Pont de Normandie between Le Havre and Honfleur. Toll is 5€ for a car. A popular site in itself, the bridge, which opened in 1995, at the time was the longest cable-stayed bridge in the world, and had the record for the longest distance between piers; these records were lost in 1999 and 2004 respectively.
There is a free bridge further south at Tancarville, and more bridges as you go further south (where the Seine isn't as wide).
Gardens of La Ballue. The winding lanes of the Mont-Saint-Michel countryside lead to one of the most magnificent gardens in twenty-first century France. The Gardens of La Ballue, with their many “outdoor rooms,” linked by stunningly designed vistas and spaces filled with imaginative topiary, recall the Italian mannerist style and the Yin Yang inspirations of the landscape architects and creators Paul Maymont & François Hebert Stevens. Many different creative ideas have been implemented, each of which has added to the gardens’ mystery and power. The design is dramatic, distinguished by three-dimensional, organic shapes and groups of sculpted allegorical figures in movement. Variety and whimsy characterise the baroque garden, and there are spaces and surprises to discover here in all seasons. edit
France is a pretty safe country, and Normandy doesn't have any big cities with no-go areas, although as in any place, you should stick to a few obvious rules (don't walk down dark alleyways at 4 in the morning, etc...).
If in trouble, speak to a policeman (Policier or Gendarme) or go to a police station (Comissariat) where you will be given help.
For health issues, go see a doctor (médecin, around 20€). For ER/A&E, ask for Urgences. You can call SOS Médecin (Tél: 36 24), who can send out a doctor (very useful in rural areas). Chemist are Pharmacies, and most major towns will have a Pharmacien de garde who will stay open all night for emergencies (they take turns, check in the local paper to get the name and phone number).
Emergency phone numbers are:
15 for ambulances
17 for police
18 for fire service (who also serve as ambulances and deal with issues such as gas leaks, traffic collisions, etc...)