In my cover letter I have the following sentence:
Furthermore I have excellent research skills and I can famililarize myself fast with new topics.
I am not sure where to put the fast in. Is this correct? Should I use fast or quick?
You should use neither fast nor quick. I would instead suggest using quickly:
... and I can familiarize myself with new topics quickly.
To modify the word familiarize, you need to use an adverb, either ‘fast’ or ‘quickly’. ‘Fast’ is a more abrupt word and, morover, can also mean firmly, so you need to be careful when using it. In almost all cases that don't involve actual motion, ‘quickly’ is the better choice for adverb.
Here are your options:
...I can familiarize myself with new topics fast.
...I can quickly familiarize myself with new topics.
...I can familiarize myself quickly with new topics.
...I can familiarize myself with new topics quickly.
The second option (2) has the nicest flow as nothing is interrupted and the flow is even. This is the one I would use if the sentence was on its own.
The last one (4) may be the most meaningful as it tacks speed onto the ability statement without dominating or interrupting it. There is also a sense of finality when reading the word ‘quickly’. This is the one I would use in your sentence.