Ask your mentors, program director and other faculty members to involve you in journal reviews early. This will allow you to learn how to appraise a paper and will provide a great learning opportunity. Of course, be certain to abide by appropriate conduct of research, and make sure the editors know you are partaking in the review process. JPIDS has the opportunity for a fellow to serve on its editorial board. Link to this opportunity is below. https://pidsfoundation.org/research/apprenticeship-program.html
Other editorial fellowship opportunities: https://sites.jamanetwork.com/fishbein
Some pointers on peer review: https://files.eric.ed.gov/fulltext/EJ1174007.pdf
See the reviewer checklist in appendix IV