This study explores the influence of teaching methods, task complexity, and user characteristics on perceptions of teachable robots. Analysis of responses from 138 participants reveals that both Teaching with Evaluative Feedback and Teaching through Preferences were perceived as equally user-friendly and easier to use compared to the non-interactive condition. Additionally, Teaching with Evaluative Feedback enhanced robot responsiveness, while Teaching with Preferences yielded results similar to the passive Download condition, suggesting that the degree of interactivity and human guidance in the former may not substantially impact user perceptions. Personality traits, particularly extraversion and intellect, shape teaching method preferences. Task complexity influenced the perceived anthropomorphism, control, and responsiveness of the robot. Notably, the classification task led to higher anthropomorphism, control, and responsiveness scores. Our findings emphasise the importance of task design and the need of tailoring teaching methods to the user's personality to optimise human-robot interactions, particularly in educational contexts. Project website: https: //sites.google.com/view/teachable-robots.