Cryptanalysis is not a standardised process with recipes to decrypt encoded messages...it has a lot to do with circumstances such as the amount (length) of ciphertext available, in what context it was retrieved (is it likely to be part of an authentication protocol, or human communication) and so forth. It often involves creativity, and lots of trial-error and patience since it is obviously a task that is difficult by design.
If there is absolutely no context information, the analyst has little choices but to closely examine the ciphertext for any kind of patterns/structure, and identify anomalies to work on. This could include things like
- Frequency of characters
- Appearances of combinations (which might originate from the same plaintext words)
- Ciphertext structure (especially if several messages have been received): do there appear to be checksums, is the length fixed,
- etc.
Note that the above is by no means exhaustive nor very helpful for solid encryption schemes. It is just to address your question and point out why it can not be answered - there simply is no standard procedure.
More importantly, most encryption nowadays does not rely on secrecy of the algorithm. Cryptographic schemes can and should be publicly known, so that the public can point out and correct potential weaknesses, which increases resource asymmetry between the "defenders" (developers) and adversaries, and thus makes attacks less likely. This ultimately leads to more public trust and acceptance of the method.
So it is no problem at all when a cryptanalyst knows that you used RSA etc. to encrypt your message, it is still infeasible to "break" the ciphertext, as you put it.
