Peer review is a cornerstone of academic communication, critical for ensuring the quality and credibility of research. Developing peer review skills sharpens evaluative abilities, promotes critical thinking, and improves teaching, making these skills an essential component of every academic's toolkit. This study reveals that, despite their importance, formal training in peer review skills in higher education is scarce. It identifies key challenges faced by novices and experienced peer reviewers and offers practical suggestions for teaching peer review skills. Quantitative analyses of rating responses from international students and researchers with bachelor's, master's, and PhD degrees were combined with a content analysis of free-text responses. Findings suggest that participants' peer review skills and motivation to write peer reviews would benefit from closer author-reviewer dialogues and iterative feedback loops. Free-text responses reveal common concerns about the accuracy and objectivity of feedback, including uncertainties in assessing research quality, judging impact, avoiding bias, striking the right tone, and communicating clearly. These challenges can be addressed through comprehensive peer review guidelines, including detailed instructions, simple rubrics, and recommendations for language use. Based on these insights, this study provides suggestions for integrating peer review practice into teaching, optimally preparing students for the assessment and evaluation demands of academic and professional environments.