Is there a term for being aware of new things, such as 'my ____(being aware of new) for music has increased since having used this new website'.
You may use awareness or knowledge,
- My awareness in music has increased since having used this new website
- My knowledge of music has increased since having used this new website
- I am up to date about music since, I started using this website.
Trendiness: the quality of being up to date or very fashionable.
The word "trendiness" comes from "trendy" meaning very fashionable or up to date.
The source is Oxford online dictionary.
I think receptivity may suggest the idea of being aware and open to new things:
open to arguments, ideas, or change; "receptive to reason and the logic of facts."
capable of or qualified for receiving.
- ready or willing to receive favorably: receptive to their proposals.
Awareness is good in the exact same context as your sentence; you could also try consciousness, such as
I've been more music-conscious since having used the website.
I've been more conscious of music since having used the website.
I [personally] would use adaptation, though everyone else's comments and answers are equally valid.
Awakening, you might have to change the wording a bit but I think it works in context.