Skill-based reinforcement learning (SBRL) enables rapid adaptation in sparse reward environments through pre-trained skill-conditional policies. Effective skill learning requires maximizing both exploration and skill diversity. In this paper, we propose Adaptive Multi-objective Projection for balancing Exploration and Skill Diversification (AMPED) to simultaneously address both exploration and skill diversity. AMPED uses gradient surgical projection to balance exploration and skill diversity gradients during pre-training, and a skill selector during fine-tuning to select appropriate skills for downstream tasks. AMPED outperforms the SBRL baseline on various benchmarks, and component-wise role analysis confirms the performance contributions of each component in AMPED. Furthermore, we present theoretical and empirical evidence that using a greedy skill selector reduces fine-tuning sample complexity as skill diversity increases.