First, sorry that your first experience is with a poorly organized site. Like any powerful tool Drupal can do some really annoying things when poorly setup, and it's frustrating when you start with a messy project.
A page may be pulling in content from a node, a view, a form, a block, or some other source. Depending on how careful the original theme developer was, for any page not providing an edit tab you may find there is a contextual menu (or several) hiding under a light grey gear icon (usually at the top right of the element is supports). Those will often provide hints about where the content is coming from, or even direct links to edit it.
It can also be useful to look over the list of views on the site (Structure --> Views), which is Drupal's query builder. The displays frequently have headers and footers defined along side the query.
After that you'll probably want to figure out what is used to organize elements on the site's pages. Drupal 7 provides several tools to do that (sometimes used together which can cause confusion) check the modules list for Panels, Display Suite, and Context. Whichever is enabled you'll want to research a bit. Each can help pull content into a page, Panels may even be setup to provide some of that content itself.