Formal:
* Curious
* Inquiring
* Investigative
* Prying (can have a slightly negative connotation)
* Interrogative
Informal:
* Nosy (can be negative)
* Curious
* Interested
* Eager to learn
* Questioning
More specific:
* Analytical
* Investigatory
* Scrutinizing
* Probing
The best synonym will depend on the specific context and the tone you want to set.