In the problem space, I often use these methods to understand and empathize:
Structured and open-ended user interviews
Stakeholder/expert interviews
Surveys
Card sorting
User journeys, storyboards, and personas
Five why's
Problem reframing
Alignment and prioritization workshops
In the solution space, I use these techniques to Ideate and refine:
Whiteboarding and Brainstorming
Wireframing
Prototyping
Moderated and unmoderated user testing
Tree testing, first click testing, and preference testing
Persona walkthroughs
Design critiques