What is a word for changing yourself to reflect the changing world? I'm looking for something else other than update and its synonyms, since they don't quite fully seem to explain what I want to say.
There are actually dozens of possibilities, depending on what, exactly, you want to say.
For example, there is FumbleFingers' suggestion of adapt.
The video game giant Nintendo started as a playing card company, but quickly adapted to the growing interest in computer-based games.
Or Canadian Yankee's suggestion of evolve (which indicates a more gradual transition)
The attitudes toward solar energy are constantly evolving. What used to be seen as an environmentally-friendly-but-uneconomical choice for the fringe is now seen as a smart long-term investment.
Other possibilities, any of which can work depending on what you want to say:
acclimate (slowly get used to something)
accommodate (to make a change in order to satisfy some external condition)
conform (change to match what others are doing)
revise (make a change to something existing, in order to correct for an error or other factors)
tailor (change to make "fit" a particular situation or use)
habituate (slowly become comfortable with something, so that it feels natural)