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A good place to start is by seeing if your site passes an automated audit using auditing tools. I'm not sure what your role will be in creating the site, or what your budget is going to be, but both Deque and Google offer free browser plugins for checking pages.

Deque's browser extension

Chrome's auditing tool

The New York Times' homepage fails AA accessibility on a number of counts, including having a non-colour hover- and focus-state indicator for links. If they'd just kept the default behaviour for links (underline on hover/focus), those linked titles would be accessible.

Beyond automated tools, you'll likely need to pay someone with some expertise. The thing about accessibility is that you can't use automation for everything - to give a simple example, automated testing can tell if you've got alternative text for an image, but it can't tell if you've got accurate alternative text.

That being said, automated testing will get you most of the way there, most of the time.