E.g if a saying or a quote is very inspirational, can we use it resonates with me? Or could someone please suggest more words for moved or inspirational?
Here is the definition of "resonate" by the Merriam-Webster Dictionary:
: to continue to produce a loud, clear, deep sound for a long time
: to have particular meaning or importance for someone : to affect or appeal to someone in a personal or emotional way
Therefore, according to that dictionary, your use of "resonate" is correct. I think you found the perfect word, though: "inspire". "Resonate" and "move" aren't very specific about how it affects you. "Inspire", however, is a little more specific:
: to make (someone) want to do something : to give (someone) an idea about what to do or create
You might also like inspirit:
: to fill with spirit
It's similar to "inspire" and is in fact listed as a synonym.
If it doesn't send you to new things, but helps you in the things you already know, you might say it encourages you:
: to make (someone) more determined, hopeful, or confident
The underlying metaphor is that of the "sympathetic string" on a musical instrument, which vibrates when another string is plucked or strummed.
So, when you say that something resonates with you, you're saying that it is sympathetic with your own experience and outlook. It is particularly meaningful for you and you may be moved emotionally by it. But you might not be moved to action, either to do something or to create something. So resonance need not involve inspiration.
I would agree with @zondo and say that "inspire" is the verb you are looking for if you mean that something has motivated you to take action of some kind, to phone a friend you have not spoken to in a long time, to take a trip, to write a poem or song, to join a political movement, to choose a career, whatever it may be.