Learning to write formal mathematical proofs presents a major challenge to undergraduates. Students who have succeeded in algorithm-intensive courses such as calculus often find the abstract logic and nonprocedural nature of proof writing to be technically difficult, ambiguous and "filled" with potential errors and misconceptions. This mixed-methods study examines 23 undergraduate students' attempts to write one-to-one and onto proofs in an introductory abstract algebra course. Data collected consisted of six rounds of assessments on one-to-one and onto proofs, including homework, quizzes and exams. Using an existing framework of undergraduate proof writing, the researchers found that students' misconceptions and errors varied substantially by student and task, with one-to-one proofs presenting unique challenges. Implications for teaching and research include emphasis on the logic of proof approaches and providing structured proof frameworks to assist undergraduates with the procedural and conceptual challenges in learning to write proofs.