How do I add Minecraft as a firewall exception?

If it is a firewall issue, which it very much sounds like the case is, then you need to add your current running Java binary to the firewall. By default, it is the following:

C:\Program Files\Java\jre1.8.0_31\bin\javaw.exe

Or

C:\Program Files (x86)\Java\jre1.8.0_31\bin\javaw.exe

The first pathway assumes you're on a 32-bit system, or that you're on a 64-bit system with a 64-bit install of Java. The second pathway is for 64-bit systems running 32-bit installs. Both pathways assume that you have the latest version of Java (as of this post) which is Java 1.8 update 31.

You can add this exception to the firewall by typing into the Windows Start search area "Allow Firewall" and clicking the "Allow an App through Windows Firewall" result, then clicking the "Change Settings" button at the top, clicking the "Allow Another App..." button at the bottom, and browsing for the binary in the above pathways. Click "Add", "OK" and you should be set.