Step 1 - Take Inventory
Though you will definitely see problems that you'll want to fix right away, you should try to hold off until you know exactly what you're dealing with. Look around the filesystem to see how the site structure is laid out. Make sure you've analyzed your web log file, have a firm understanding of your traffic patterns and know the site's audience.
Having a solid baseline is an extremely important first step since this baseline is what will allow you to measure the impact of the changes you're about to make. Your baseline is also important because as you process the information needed to get a baseline picture of your website, you may find some surprising elements to the site. Every site has its own traffic quirks, be sure you find out what they are on your new site.
While you're taking inventory try to get a good grasp of where you are with respect to the search engines. Are the robots crawling the entire site? What are your top referring search phrases? Which search engines are sending you the most traffic?
Step 2 - Consider Automation
If the site you're trying to improve is large, then automation is something that's going to make your life much easier looking into the future. When people hear the word automation with reference to websites, the first thing that normally comes to mind is a content management system A content management system is a significant step along the path towards automating your website, but it's not the only kind of tool available.
One of the tools you can incorporate into your automation scheme is a custom robot that will crawl your site looking for items of interest. Those items of interest could be broken links, pages with validation problems, spelling mistakes and anything else that could be of use to you and your work.
If you put together this kind of automation early on you'll find many of the later tasks easier. You'll also have the tools to make the long term maintenance of the site quite manageable.