The FRG assumed the legal point of view that there was only one German citizenship, and thus recognised citizens of the GDR as its own citizens.Until 1969, the FRG did not recognise the GDR as a country, but as basically occupied territory, and insisted on its exclusive mandate to represent the German people; relations thawed somewhat to the point of acknowledging two countries in Germany, but citizenship of the GDR was still recognised automatically as FRG citizenship (as a side note, up until the 90s, Germany - seeing itself as the souvereign representation of the German people, had a very lax immigration and naturalisation policy for ethnic Germans in general).

As for crossing into the FRG, it must be noted that the border wasn't immediately fortified in 1945, so irregular border crossings between the Allied zones and the Soviet zones remained possible.
The general border wasn't particularly fortified until 1952, afterwards the majority of people fled across the sector border in Berlin. Here's an online available statistic sourced to an official paper by the state of Berlin; the first table shows escapes until the building of the Berlin Wall (year - escapes via West Berlin - escapes via the rest of the border - total); the second shows the successful escapes via the Inner German border after the Wall was build (this section of the respective Wikipedia article, sourced to the Ministry of Intra-German Relations, shows total refugees vs this number).

In West Berlin, refugees were registered at a central transit camp and then flown out to the rest of West Germany (like other refugees, they were not free to move but actually distributed among the different states). So obviously, this influx was very much reduced after the Wall was built in 1961.

Afterwards, people did try to leave directly for the FRG in a number of ways; including tunnels, boats or swimming across the Baltic Sea, ballons, aircraft, ramming the border with cars, smuggling in cars, and in one instance, an aircraft hijacking. You can find some photo documentation here.

The majority of post-1961 refugees fled via a stay in another country, however; either trying to cross another border, or defecting when they got permission to go on a trip into non-COMECON countries.