How can I keep up with all the important advances in fields related to mine?
There is no easy way:

First step would be to talk to your advisor.

Second step would be to look at recent issues of the main journals in your field.

Third step would be to attend some of the main conferences in your field, listen to the talks and talk to the people.

After that you make up your mind what your opinion the trending issues are, and repeat steps 1 and 2 (and maybe 3) to try to confirm that.