change solution file to a different folder

The solution file is just a text file. You should be able to move it and then edit it to make sure that any relative paths referenced in the file are correct.

I've also been able to right-click on the solution file in the solution explorer and then use save-as to save it in a different location. That doesn't seem to work everytime for me though.

Open solution explorer in Visual Studio and select the solution.
Click on File > Save MyProject.sln As ...
and choose the new location.

Open a blank solution, save it to a location where you would like to move. Then add "Existing Project" and select the Old.sln file to add all the projects under the old solution file.