How do I get to the next search match result, or perform a search and replace?

If you first click in the source code panel and then Ctrl-F (or Cmd-F if on a Mac), that should bring up a search-and-replace dialog box. You can use the previous/next match buttons to move through the search results.

These functions can also be accessed via keyboard shortcuts — you can find a list of all shortcuts available on Overleaf here.