It's not expected (or remotely possible!) to be perfectly up-to-date on everything being published, but some active effort is better than none.
It's crucial to keep up-to-date with the academic literature. Generally, it is recommended that you set up a Google Scholar alert for authors and keywords related to your research. You may also consider subscribing to a journal's RSS feed with an RSS reader app. Reeder seems to be a popular tool for Apple products.
For literature searches on previously published work:
Explore the literature through citation networks via tools like ResearchRabbit. Even more simply, use Google Scholar to see which papers cite a paper of interest and use that to broaden your search.
Remote Access
The easiest way to access journal articles off-campus:
New journals keep appearing, so this list will surely be out-of-date soon. That said, here is an incomplete list of journals where we often find relevant work:
General science journals:
Science
Nature
Nature Communications
Science Advances
General chemistry journals:
Nature Chemistry
Journal of the American Chemical Society
ACS Central Science
Angewandte Chemie International Edition
Chemical Science
JACS Au
Computational chemistry and computational materials science: