Download and install MiKTeX first.
While one can edit LaTeX source files (.tex files) with any text editor, it is helpful to use an editor that is aware that it is editing LaTeX, one that marks the different portions of the text according to their meaning in LaTeX, one that makes it easier to run latex and pdflatex, and one that can help you when you don't know the LaTeX code for something. It is also good if the program is available free of charge. LEd is one such editor, and I have been happy with how it works. Perhaps the best thing about it is that it automatically locates your installation of latex, so you don't need to tell it where it find it.
You can read about the LaTeX Editor at the LaTeX Editor webpage.
To install it, start at the download page. To use LEd in a computer lab, it is easiest to download it in zipped format, see the link below:

Right click the link (in blue, it currently reads 5922 KB), then Save Link As, and direct it to a sensible place on your computer, like the C:\ folder or the My Documents folder. Once it is done downloading, navigate to the .zip file, right click it, and choose Extract to a folder. Open that folder and double click on LEd.exe:

LEd will ask you if you want to associate certain file types with LEd and latex:

I think it is safe to say Yes, all, but it's your choice. The main ones are the .tex files anyway. This way, when you double click a .tex file, it will either open LEd or run a version of latex on it. You can start LEd by right clicking a .tex file and choosing to Open With LEd.
If you are the computer administrator, you may want to really install LEd on your computer. To do that, choose the Installer on the download page and follow the instructions. The advantage is that LEd will create a folder under Start, Programs so that it is easier to start LEd.

The PCs in the math building and Hayes 025 don't know where the program latex.exe or pdflatex.exe are. Here is one way to deal with it:
Open the folder “c:\program files\LEd\batches”
Right click and edit the file LaTeX.bat
Add the line:
path "c:\program files\miktex 2.9\miktex\bin”
before the line that starts "latex.exe" Modify this as appropriate for your installation of MiKTeX.
First, figure out exactly where the MiKTeX program files are installed. Probably in C:\Program Files\MiKTeX 2.8\miktex\bin, or 2.9 if you have the more recent version. Click in the address bar to prepare to copy the "path" to this folder.

Now, where to paste it? Start, Settings, Control Panel, System:

Click the Advanced tab:

and then click Environment Variables. In the bottom window, highlight the line that starts "Path":

The Path variable tells the PC where to look for executable program files. Click Edit. This will give you a very little dialog box.

Very carefully paste in the path to the MiKTeX bin files, separating this new path from previous paths with a semicolon (;). Say OK and then restart LEd.