When you use Nautilus, just click search from the top level of your home folder (i.e. where you can see all your folders laid out) and in the search box only enter .pdf (or whatever extension like .txt that you want to find). That's what I do and it just returned me all the pdfs in my home folder. You don't need to use a wildcard or put quotes around the search term in Nautilus search, unlike when you use search programs on the command line.

The gnome-search-tool can be used to search within files- select the home folder or the directory to be searched and then click select more options, and input your search term where it says 'contains the text.' It can be quite slow as it works without an index, but I have used it several times and it has been useful for basic searches.