I'll start with auditing the website.

Create a staging site if their isn't any

Apply the recommended optimization techniques, which include but are not limited to.

If those don't work, I'll dive a bit deeper and modify and optimize the code.

I'll use a software to check the site health, and solve any issues that reside from it, that will by extension improve the page speed.

When all's done, I'll apply the changes on the live site.

QA to make sure everything is working perfectly.

Hand over the website.