I am using a Samsung Galaxy S5 and I have just installed a Cyanogenmod 13 Nightly. I have TWRP and I have already installed the gaaps package. I have google play installed but it won't open when I press it, it would just pop back to the home screen. And I couldn't install any apps online right now because I am the default browser doesn't allow me to download anything. So when I tried download an apk, or anything, the browser simply closes and says "Unfortunately, Browser has stopped." Please help!
You could try to download .apk to your computer and then use
adb install your_package.apk command to install it to your device. Android SDK is required to do this.
UPD: Also, as Firelord mentioned in comment - do not use nightly builds. They are unstable. It could be a reason. Try some stable build.