Alternative spellings of the name of this state are Baden-Wuerttemberg and Baden-Wurttemberg.
Among the West-German states, Baden-Württemberg is one of the youngest, having been founded in 1952 through a unification of administrative areas that, until the end of WW I in 1918, had been mostly covered by the kingdom of Württemberg, the grand-duchy of Baden and the kingdom of Hohenzollern. The consequence of this - and that's the important bit a traveller should know - is that there are now two tribes living together in the state: Badener in the west and Schwaben in the east. Both speak different dialects (see below) and share a love-hate relationship towards each other that's nurtured with a lot of humour. For what unites both tribes and the rest of the people living here is a pride for "their" Baden-Württemberg and what they have made of it since its creation, that's surprising for Germans from up north. Since 1999, the state has been advertising itself all over Germany with the slogan "We can do everything, except for speaking Standard German." (Wir können alles, außer Hochdeutsch), a tongue-in-cheek play on the infamous dialects (see below).
And indeed, Baden-Württemberg is doing quite well in terms of economics compared to other places in Germany. It boasts the lowest unemployment rate of the Federation, some of the best universities in Germany, a GDP per capita that rivals Switzerland and is the only German state that still has a higher birth than death rate. The European Statistics Office (Eurostat) has called Baden-Württemberg the "high-tech central of Europe". And, famously, the percentage of people owning their own home is by far the highest in Germany.
The main reason for all those superlatives lies deeply in the history of the land: Although nowadays there are about as many non-Catholics as Catholics living in Baden-Württemberg (and a third group of comparable size without religous faith), during the reformation South-West Germany was strongly influenced by the schools of Martin Luther, John Calvin and Huldrych Zwingli, which left behind a society with moral values circling around hard work, self-control and the general motto "God helps those who help themselves".
Hence the country that was once dirt poor, having to struggle with hard winters and frequent famines, today is plastered with high technology companies. The most important sectors are mechanical engineering (most famously Robert Bosch Inc.), Chemistry, Biotechnology and, above all, Automobiles (which were, in fact, invented here, as everyone will be happy to point out). Daimler and Porsche were founded and still have their headquarters around Stuttgart; Audi, Volkswagen and others have large plants in the state. If one counts in the small and medium-sized suppliers, every other employee in Baden-Württemberg is working for the car industry, directly or indirectly. As Max Weber, a philosopher at Heidelberg University said, around here, it's "Capitalism as it was meant to be".
While every region in Germany has its own Germanic "dialect" in addition to Standard German (Hochdeutsch) Baden-Württemberg (together with parts of Bavaria and Saxony) is among those regions where the "dialect" is actually the native language of the near-majority of the population (except in the north).
The traditional "dialect" in most of the state is Alemannic (Alemannisch) which is by far the main language in German-speaking Switzerland, Liechteinstein and Vorarlberg in Austria, as well as being spoken natively by many is western Bavaria and as a minority language in Alsace in eastern France. As it is divided into numerous local dialects and has its own written language, it is very disputed as to whether it is a dialect or in fact a separate language. More and more people understandably state the latter.
The exact proportion between native speakers of Standard German and Alemannic is unclear; however in general more Alemannic speakers are found in rural areas than in say, Stuttgart, where Standard German nowadays seems to be the more common mother tongue.
Kurpfälzisch is the traditional language in the north of the state (i.e. the region surrounding Mannheim and Heidelberg) but standard German is what dominates in most places. That said, it is still spoken by many people in the rural areas.
Many Alemannic-speakers are fluent in Standard German and in English, even in rural areas, but are also quite proud of their "dialect". Visitors are encouraged to learn a few words or phrases in Alemannic to get the full experience.. Although native Standard German-speakers are a majority in many cities, you still will encounter plenty of native Alemannic-speakers as well, some of whom might in fact be uneasy about speaking Standard German (mostly rural elders).
All in all though, language is not a major barrier, and even a monolingual English-speaker should have no difficulty truly enjoying this sunny part of Germany.
Stuttgart has an international airport which is served by all major carriers. Frankfurt international (FRA), the busiest airport in mainland Europe, although not in Baden Württemberg, is well within reach by train (1 hour from FRA to Stuttgart main station via the high-speed ICE connection). Low-fare airlines offer services to the local airports of Karlsruhe-Baden Baden and Friedrichshafen.
Travellers beware: "Frankfurt Hahn", the big hub for low-fare airlines, should not be confused with FRA. In stark contrast, it has no train station and is in a rather remote location. It is possible to get from Hahn into Baden-Württemberg rather conveniently, but it definitely takes a lot longer and is much more hassle than from FRA.
All major cities are well connected through the Deutsche Bahn rail system. Ulm, Karlsruhe, Mannheim, Heidelberg, Stuttgart and Freiburg even have ICE connections (slick, comfortable, white high speed trains travelling at up to 190 mph (300km/h)). Tickets can be booked via the Deutsche Bahn Website 
Baden-Württemberg (as well as some other regions in Germany) offers a special regional train ticket (in this case, the Baden-Württemberg ticket). It is valid for one day from 9am. With this ticket up to five people can use all regional trains within Baden-Württemberg for 21 EURO for a single ticket(Add 4 EURO per person afterwards).That means you can use all trains except InterCity(IC), InterCityExpress(ICE), EuroCity(EC) and some special trains.
By train and bus
Baden-Württemberg has an excellent rail network, serving even quite remote areas. Especially rural villages are served by buses which generally leave from main train stations in larger towns and cities. Buses are quite frequent near big cities, but especially on weekend in rural areas there are only 2–4 bus connections a day.
If you're travelling within Baden-Württemberg, you can purchase the Baden-Württemberg-Ticket , which will give you all-day travel in regional trains (categories S, RB, RE and IRE) within Bavaria and even to the neighboring cities of Basel, Lindau and Würzburg. You can use it also for private trains and most of local buses and city transport. On working days the ticket is valid from 9AM to 3AM the following day. On weekends is valid from 0AM.
There are five variants of Baden-Württemberg-Ticket:
For general information about Länder-Tickets see Germany#Network_tickets.
Of course you can always use your car. If you are travelling in the Black Forest or the Swabian Alb during winter, bring snow chains as some smaller roads may not be plowed frequently enough. When travelling on the Autobahn, the same precautions as everywhere on German high speed roads apply: If you're not willing (and prepared) to drive consistently above 80 mph (130 km/h), stay on the right. Make room for people trying to overtake, use your common sense, don't drive faster than you can think.
For those interested in "high culture":
For those fond of nature:
For those interested in touring castles
For those interested in tourist routes
The official tourism homepage is at http://www.tourismus-bw.de/. Click on the "English" link at the top.
Baden-Württemberg contains some of Germany's most significant wine-growing regions. Much of the wine economy is in the hands of local co-operatives and the locals enjoy the wine in old-fashioned wine cellars. The best wine grows in an area called the Kaiserstuhl in Baden.
Fruit brandies, e.g. Obstler (distilled from apples and pears) and Zwetschgenwasser (plums) are just two of the most common spirits. The queen of Schnaps is without any doubt the Kirschwasser (also sometimes referred to as Kirschwaesserle) made out of cherries from the black forest area. These are commonly drunk after a meal in a restaurant.
There are some breweries of note in the region, of which Rothaus or Welde are two beers which enjoys cult status.
Baden-Württemberg is one of the safest regions in Germany. In large cities like Mannheim and especially Stuttgart, be aware of theft. Other regions are safe and you can travel alone without any problems. Even walking alone late at night is no problem. When out hiking and trekking have a map and take proper clothing. The forests are thick and dark and surprisingly rural considering the population density of Central Europe.