What can be learned from an encrypted text? I assume nowadays crypto algorithms are well-built, so one can't make a difference between a ciphertext and random bit stream. This way the method used for encryption can't be figured out either.

I am curious, if there is perhaps something else that can be derived from a ciphertext, some meta data for example.
First lets have a look at your attack model: attacker is eave dropping only. This means no modification, just listening.

You also assume that the encryption system is perfect and you can't differentiate it from a random oracle (if I'm not mistaken we call this indistinguishability).

Due to all these restriction, if the system is well devised (no meta data). The only pieces of informations that you would be able to catch are the length of the data or the existence of the transmission (when are they communicating etc...)

Nowadays in order to avoid giving such data, (military grade) communications just send random noise when not used. That way, you won't be able to tell whether or not they are communicating. So no you cannot retrieve informations in this context.

Considering the use of meta-data, then you can retrieve some kind of "context-sensitive" pieces of informations by doing traffic analysis.
E.g. you can track communications between the hospital and your employees ... You don't know what they are about, but you know that your employee might be having some health problems...

More to read here : TRAFFIC ANALYSIS: or... encryption is not enough by Carmela Troncoso, IMDEA Software Institute.