Formal:
* Becoming acquainted with
* Establishing a rapport with
* Developing a relationship with
* Learning about
* Familiarizing oneself with
Informal:
* Hanging out with
* Bonding with
* Connecting with
* Catching up with
* Getting to know each other
* Hitting it off with
Specific Situations:
* Learning the ropes of (for a job)
* Getting the hang of (for a new skill)
* Finding out more about (for a topic)
You can also use verbs like discover, explore, understand, appreciate, or grasp to replace "getting to know" depending on the specific meaning you want to convey.
For example:
* Instead of "getting to know the city," you could say "exploring the city."
* Instead of "getting to know your new colleagues," you could say "discovering your new colleagues' personalities."
Remember to choose the phrase that best fits the tone and context of your writing or conversation.