Eastern Germany includes most of the states that used to belong to the former country of East Germany (Deutsche Demokratische Republik, DDR). The highlight is the metropolitan capital of Germany, Berlin.
The official language is German, spoken both in the form of standard German and various local dialects depending on the area. For example, there is a local dialect in Berlin but quite few native speakers remain, while in rural and to a lesser extent urban Saxony the local Saxon dialect is the majority language. Although the vast majority of Germans can speak standard German, it is for many of them not the native language, and one should note that German dialects are as different as separate languages.
While Germany is not the most English-proficient country in Europe, English-speaking Germans are still a majority, and especially younger people are often fluent. That said, German is not a difficult language to learn and Germans are very tolerant with non-native speakers so feel free to try some phrases. During communist times older people in the former East Germany had to learn Russian at school and you might find someone who is still able to speak the language, but it is increasingly uncommon. The small minority of the Sorbs is living in the rural areas at the border between the federal states of Brandenburg and Saxony. Those cities and villages are bi-lingual using both German and Sorbian, which is similar to Polish and Czech
The major airports in the region are in Berlin, Leipzig/Halle (airport code: LEJ) and Dresden. Berlin currently has two airports, Berlin-Tegel (TXL) and Berlin-Schoenefeld (SXF). Currently Schönefeld is extended to become Berlin-Brandenburg International Airport (BER), which is planned to open someday after 2014 and replace the existing airports. The extensive construction site combined with the scandals around planning and construction failures and time overrun make it a tourist sight in its own right..
Major international airports outside Eastern Germany but somewhat close would be Hamburg, Prague and Warsaw.
Germany has an extensive train network, and the area is well connected with trains from all over Germany and the bordering countries of Poland and the Czech Republic. Berlin Central Station functions as hub of train transport in Eastern Germany. The most prestigeous line of the German train company Deutsche Bahn is the high-speed line Hamburg-Berlin, where hourly InterCityExpresses can go up to 300km/h fast. Other major inter city connections connected by EuroCity-trains or InterCityExpresses would be Kopenhagen-Hamburg-Berlin-Dresden-Prague, Warsaw-Berlin, Berlin-Leipzig-Munich and Leipzig-Frankfurt/Main. Besides that there is an extensive network of regional trains servicing smaller cities.
Germany is famous for its network of motorways and since the reunion of 1990 the network has been extended and upgraded. You can get into the area from everywhere in Germany quite fast and convenient, however, the connections from neighbouring Poland or the Czech Republic are still in development and you might find a massive difference in road quality on both sides of the German border.
You can rent cars in all major airports and will find renting outlets in most bigger cities. Traveling by car is the most flexible and the network of motorways and major roads is sufficient to bring you anywhere in Eastern Germany within hours. It tends to be fairly expensive as well.
Note however that most car rentals deny having their cars taken to eastern European countries, including Poland and the Czech Republic. If you plan to visit these countries as well, you might chose to rent your car there, as those limitations do not apply the other way round.
If you have a sufficient sense of adventure and at least a small grip on the German language you can take regional trains that connect almost every small city with major hubs like Berlin, Dresden and Leipzig. There are train tickets intended only for use on regional trains, that are called Ländertickets (state tickets) and come as "Sachsen Ticket" or "Brandenburg Ticket" and are only valid on regional trains in Saxony and Saxony-Anhalt (Sachsen and Sachsen-Anhalt) or Brandendenburg, respectively. To get around by train gives you the possibility of getting in touch with the locals. Younger people and students are the most likely to speak English, so approach them for a chat and they might tell you about hidden gems worth visiting that no guidebook would ever cover.
Brandenburg and Berlin
Like the rest of Germany, the whole area of Eastern Germany is a very safe place to travel. The laws are strictly enforced, the police are easily approachable and very likely to help. In the course of the Football World Championship 2006 the police force of the major cities received enough training to even deal with foreigners in English. In smaller cities you will find the need to talk German. The emergency phone number is 112 for every type of emergency.
While large cities such as Berlin and (to a smaller extent) Leipzig and Dresden have become multi-ethnical and have large immigrant communities living there, in smaller cities and in the countryside, non-white visitors might sometimes attract wary looks. In some places there are right-wing Neo-Nazis, which may look for trouble, especially when drunk in the evenings. Act with common sense, and in doubt stick to bigger crowds of 'regular' people. Don't be afraid of calling the police when feeling uneasy. Sadly, not all hate crimes are committed by Neo-Nazis.
Beggars and Pickpockets
While the German social systems allow everyone a reasonably decent living from social welfare, bigger cities such as Berlin or Hamburg have their share of beggars and homeless on the streets. Those people are harmless and closely watched by the police. Especially in the vicinity of train stations or in crowded pedestrian zones you should be aware of pickpockets. Again, there is nothing extraordinary, and common traveller's sense will suffice.
Germany is relatively open-minded towards homosexuals. Nowadays Germany even had a gay foreign minister and the city of Berlin is ruled by a gay mayor. Current law is in a process of change and gay/lesbian couples can even have same-sex marriages with a number of fiscal rights attributed to that.
Overt display of homosexuality is relatively safe in gay neighborhoods in big cities such as Berlin. In the countryside and in dangerous neighborhoods, displays of homosexuality can attract wary looks, and with Neo-Nazis and Muslim extremists around may result in insults or worse. You are better off, applying common sense.
to the North
To the East
To the South
To the West