The difference between the two is illustrated by the following examples with a similar context.
To take up something, means to start doing it frequently, e.g.
“When I went to live in Switzerland I took up skiing, but I was never very good at it.”
(The first clause is similar to the example from definition 4a in Merriam–Webster.)
To take to something means to immediately or quickly become proficient at (and enjoy) some new thing, e.g. in the standard phrase:
“He took to skiing like a duck to water, and soon he was competing at national level.”
(The idea being that ducks do not need to be taught to swim, they do it naturally. Again — by chance — I find a similar skiing example in Merriam–Webster.)