If you have a document that doesn't allow you to highlight, underline, strikethrough, or search its text, that document has likely not undergone Optical Character Recognition (OCR). OCR is required to make document text selectable and searchable.
You can OCR a document in a desktop application like Adobe Acrobat, or by using a web-based OCR tool, such as the one found here: http://www.onlineocr.net/. In the case of documents that do allow highlighting, but inconsistently, you can usually resolve this issue by exporting a new version of the document via Preview (File > Export).
If you are unable to OCR the document in order to use the dedicated Highlight tool, you can still mark it up using other annotation tools. The Pencil tool, specifically, can be adjusted to the thickness and transparency of a highlighter using its Options menu; the Straight-Line tool can similarly be used to underline or strikeout un-selectable text. Neither of these tools require the document to have been OCR'd.
Example of Pencil tool settings for use as a freehand highlighter: