Some questions that might help with the strategy:
In what position are you working - role you were hired? What’s your experience with testing, the domain of the product?
Are you alone or are others there as well in the testing team? How much do they know about testing or about the product/project?
What’s your time? How long have you been hired to support that project? 1 week, 6 months, unlimited?
Are you located with the team or remote or offshore?
What hardware & software do you have available for testing purposes?
Can you manage the environment or do you need help from others?
What were you hired to help with?
What is your current mission? Who do you report to? How often? What information would they like to know and when?
Are there any milestones that you should consider? When are they?
Do you have access to people, code, documentations, minutes of the product discussions, tasks that have been tackled, architecture models, emails, data;
Do you know the stakeholders, or could understand what they need from such a product?
Is the product fully developed inside or using other things(systems, APIs, devices, companies, databases, etc) - how is the communication being made?
How often is the product changing? Do you get informed? Can you get informed?
Are there regular meetings about product management or features? Can you try to get in those meetings to catch the direction, meet the people, understand the inner workings…
Are there similar products on the market? What stage are they in, how do they differ or are similar, how is your product going to compete with those?
Also, some good ideas are present in the Product Elements, Project Env, Quality Criteria in James Bach’s HTSM: https://www.satisfice.com/download/heuristic-test-strategy-model