Action and ambition are synonyms for the word initiative. Sarah took action to ensure whales were properly protected on the coast.

The word initially is indeed an adverb, yes.

An example sentence is:
We were initially very excited about the trip.

The best way to improve one's ability to string a guitar is through practice. Initially working alongside a professional or a written guide can ensure proper introduction to the techniques for efficient and accurate stringing.

Initially, I didn't want to write this sentence, but I changed my mind.

Neither rigid nor initially are synonyms for worse.