Physics education research to date has predominantly focused on the undergraduate level. This has left critical avenues of research in graduate education relatively understudied and has motivated much of my research. Numerous studies across the education research landscape have analyzed undergraduate admissions practices, but few have critically examined physics graduate admissions to the same extent. Additionally, significant research has focused on identifying characteristics indicative of productive PhD mentoring relationships, yet no study in physics has investigated how graduate students go about finding their advisor and research group in the first place. Lastly, despite ample research on problem-solving in undergraduate physics classes, little work has examined the expert-like problem-solving skills typically used by graduate students during research. My work addresses these gaps in the literature. First, I present results from a quantitative study of admissions data from 1,955 physics graduate students to illustrate how undergraduate grades predict graduate grades and PhD completion more effectively than GRE scores. Counterfactual mediation analysis is used to show that undergraduate grades predict PhD completion indirectly through graduate grades. I then present a qualitative study investigating 40 first and second-year physics PhD students' perspectives on the experience of finding a research group. Results from this study indicate that helping students successfully navigate the group search process may positively impact retention through increased sense of belonging in their programs. However, students often perceive a lack of guidance from their department regarding how to navigate that process. I also discuss the role that research interests play in the group search, including their effects on when students join a group and the concerns they have during the process. The final study details results from interviews with 11 theoretical physicists that give insight into how making assumptions and using analogies are used in their professional research. I discuss connections between the ways theorists use assumption and analogy and offer potential new avenues of research regarding applications to instruction at the undergraduate and graduate levels. I conclude by synthesizing the results of these studies and their implications for making PhD programs more inclusive, supportive, and productive for all students. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://bibliotheek.ehb.be:2222/en-US/products/dissertations/individuals.shtml.]