To "take up" the use of grammar means that A) he is suddenly interested in it or engaged with it.

The narrator notes that once he begins paying attention (or engaging in the correct use of) grammar, it becomes difficult to even speak Italian until he processes everything correctly in his mind.

To examine the other (incorrect) things "take up" could mean, the narrator isn't beginning a new position/job, so that context doesn't fit. He's just paying attention to the rules governing usage of Italian speech. He isn't really occupying a time or space here that is conveyed in this phrase. That would be something like, "I took up half the aisle with my bags and buggy." And he is not saying that he's going to postpone his knowledge of Italian grammar, which would be something like, "I decided to take up Italian grammar later."