The way you construct your contract is different from the approach to development (agile, waterfall, etc.). On the topic of it being fixed or time and materials, I always pick a fixed / delivery-based contract. Especially if this is your first big project and you are non-technical. When you select the agency-developers, you want them to put together an SOW that has as much detail in it as possible. Vague terms can be misinterpreted. "Build front page" To you, that can mean so many things, like the special carousel calendar you mentioned in a meeting, but unless it's spelled out in the SOW, don't expect it to happen. If the project is taking longer than expected, it's still on them to deliver what is promised, as opposed to a T&M-based contract... where they can take longer and keep charging you since you're paying for their time. If they won't do a deliver-based agreement, then there are other ways to get what you want in the SOW with less risk to you, but it's tougher.
The development approach largely depends on the people involved and your timeline, but most often I agree that going agile is best since you see releases sooner and by definition is more accepting of changes later in the process (something that might be key for a first project).
Curious, is this a business that is centered around the website? Is the product you're selling the website? If so, I'd say find a great developer and build it in-house. Under this model, there are some investors that won't fund companies that didn't build their own product. Which is sound investing, but there are good reasons you want to build your own product.
Good luck and feel free to reach out for any follow up questions.