Well, .tar.tar doesn't really make much sense. That would be ineffective, are you sure that is what it is?
.tar.gz is a compressed tar archive. 90% of the time this is going to contain an application in source code form, which means you need to compile it once you have extracted it. To open such a file you would do something like:
tar xvf ./filename.tar
Note that the .gz extension is removed after the first command.
Now, the .sh files are executable installers. For those you would first make them executable, and then run them. Like so:
chmod +x ./filename.sh
Then it would likely start an interactive installer of some sort, depending on what it actually does.