Key topics for tutoring a beginner in algebra include focusing on the order of operations, working with negative numbers and fractions, and simplifying algebraic expressions. It's recommended to start with exercises from the student's textbook for familiarity, but alternative resources can be used if the textbook is unhelpful. A six-minute revision test at the beginning of each session can be beneficial for tracking progress. Reinforcing the methods taught in the student's class is crucial to avoid confusion, while adapting teaching strategies based on the student's understanding is essential. Overall, the initial tutoring session should assess the student's current knowledge and build from there.