Start with the business problem, stated as a measurable result. "We need a new website" describes a purchase. "Demo requests from the site have been flat for a year while traffic grew" describes a problem, and it gives the agency something to investigate.
The goal carries extra weight in this method because later work is chosen with it in view. The strategy stage begins with clear objectives and an understanding of what customers need, then turns a wishlist of ideas into a plan. After launch, each improvement cycle starts by choosing one metric to improve. An agency has little to rank ideas against, and little basis for choosing that metric, before it knows which result you're paying for.
Next comes what you know about your customers and how you know it. Sales call notes, support tickets and customer interviews are evidence. A persona built only from the team's opinions is a set of assumptions until someone checks it against real buyers. The agency should ask which of these you have, because the answer affects how much research the paid strategy has to include.
Technical constraints belong in the first conversation too. That means the CMS the site runs on, the CRM it feeds, the integrations that have to keep working, and how much content would need to move. Platform alone can change the scope. Lean Labs, for example, says its growth-driven design program is built specifically for HubSpot CMS, and that it can migrate a company to HubSpot as part of the engagement. A buyer on another platform needs to hear that on the first call, since it adds a migration to the work.
Then there's the question of who decides. Lean Labs' guide to choosing a growth-driven design agency recommends a single person who has final say and controls the budget, backed by one or two people who know the product and the customers. It also expects an experienced agency to raise this before any contract is signed. Ongoing work runs in short cycles, so a change that waits on several approvals can slip past the cycle it was planned for.
Budget belongs in the conversation as well. Give a range, and ask what the strategy, the first build and the work after launch would each cover and on what terms.
Expect the agency to ask a good deal. Much of what it needs at this point is in your head and your analytics.