I wonder how to say something like
Now I'm reading about ways to get to Kyoto from Tokyo.
I know the necessary words but I wonder how to say this "about". Will it be okay if I just write this?
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Your sentence is grammatically correct. について is a good way to say "about" in many situations. However, in this case I think it would be more natural to say
I'm [ looking up / reading about ] how to get from Tokyo to Kyoto.
調べる is often translated to "investigate, examine", but it's a very common word and I think "look up, read about" are equally valid translations.
について読む sounds a bit like you are reading, say, a monograph on historical travel routes.