I would recommend you guys hire an Agile Coach or Scrum Master, at least for a six months period. One of the four parts of the agile manifesto is Individuals and interactions over processes and tools so answering your question, there isn't a right or wrong way of doing thing but a way that the whole team agree on following.
Just to get the ball rolling, let's look at your question through a tunnel. So you are redesigning a site and have decided to switch to agile, for example Scrum.
First and foremost you will need a Product Owner. This person will be someone who understands the business goals of the redesign project. Maybe it is better conversion or perhaps a CMS to allow different staff to update the pages/data without relying on the technical team.
Second, you will need a development team. This will include yourself.
Thirdly, you will need a Scrum Master, who will ensure Scrum framework (not process) is in place and helps rest of your organization and maybe even external stakeholders understand what being agile/lean means.
Now that we have the team in place, let's decide how to get things done.
The Product Owner will need to prepare a Product Backlog. This will have the business requirements written as "User Stories" and prioritized by business value; i.e. the most important requirements will be on top of the list.
So when the homepage, page x or page y is done first really is at the discretion of the Product Owner. Who in turn must both represent the business stakeholders and the top management, as well as make decisions based on the best interest of the business.
A final note on this though. Although the Product Owner decides what requirements are more important, the development team and only them can decide how much work they can get done in a sprint, which is a 2 to 4 weeks period that results in a software increment. The product owner, development team and scrum master work as one team. Together, they are responsible for the codebase and the expected results. They fall together and rise together. The same is true within the development team, which is cross-functional and must work as one unit. Imagine a an army unit at battle or a football team made up of different skills but all working in synergy to achieve one thing!
Scrum as a framework is very simple to understand but is exceptionally tricky to execute, especially when an organization is used to working in the waterfall way. All change takes time and this is no different, hence the importance of hiring a qualified Scrum Master or ensure your existing project manager is excited to become a servant-leader of a self-organized team.
Hope this helps.