Analysing the ERP (Event-Related Potential from EEG recordings) in terms of the P-300 wave
So, are repetitions wanted in this kind of experiment?


Yes: repetitions reduce random noise and increase the chance to tell artifact from response. Adaptation is a serious issue, typically becoming apparent by decreasing amplitudes, or even skipped responses. Reading the existing literature and/or performing experiments with various stimulus intervals may help to identify adaptation effects, if any.


  ...there is allways a background activity in the brain...


Correct - if not, you are either not recording properly, or subject is brain dead.


  are there certain frequencys that are more likey to happen under these conditions  


If you repeat your recordings and average them out, this is not an issue. Otherwise, yes, FFT analysis shows several frequency bands, dependent on the state of the subject. For example, awake subjects show primarily fast beta activity (13-30 Hz), drowsy or sleeping subjects much slow-wave alpha (1-4 Hz). 


  should I assume anything on the P300 response?


In short: The P300 response is most commonly elicited in an "oddball" paradigm when a subject detects an occasional "target" stimulus in a regular train of standard stimuli. The P300 wave only occurs if the subject is actively engaged in the task of detecting targets (Picton, 1992). 


  What is the recommended sample Rate I should use. I guess there should be an advantage for high sample rate for more resolution bu[t] I am not sure about that. Am I right, or maybe a lower samp[le] rate has advantages th[a]t I am not aware of?


The higher the sample rate, the better. Downsampling can always be done later. Typically 1000 Hz is sufficient for fast responses. For a P300 this should already be kind of an overkill.

Reference
- Picton, J Clin Neurophysiol (1992); 9(4): 456-79