Learn Russian: По ознакомлении с проектом мы немедленно приступили к работе. - Upon familiarization with the project, we immediately proceeded to work.

The preposition по meaning "upon" or "immediately after" requires the Prepositional case. Since the dictionary form of the noun is ознакомление (a neuter noun ending in -ие), its Prepositional case ending is -ии. This structure is highly characteristic of formal Russian.

Could we just say после ознакомления instead?

Yes, после ознакомления (using the Genitive case) means the exact same thing and is much more common in everyday speech. По ознакомлении is a formal, bookish construction mostly used in official documents, business correspondence, or literature.

Why is к работе in the Dative case?

The perfective verb приступить (to proceed to, to start) always requires the preposition к followed by the Dative case to indicate the task you are starting. It translates literally as something close to "stepped up to the work."

Can we use an infinitive after приступили, such as приступили работать?

No, you cannot. Unlike начать (to begin), which easily takes an infinitive like начали работать, the verb приступить requires a noun phrase with к. If you want to express starting an action with this verb, you must use a verbal noun, such as приступили к чтению (proceeded to reading).