So you've found a route here on OpenRoutes that you'd like to follow, but you're unsure what you're supposed to do with the GPX file you've downloaded. Here's my guide to get you started.
First, let's make sure you've downloaded the route. On each route page, you'll find a link to download the route, like this:
Right-click on the blue link and (depending on your browser), select 'Save link as...' or 'Save target as...' or something similar. Select a folder to save the file to (e.g. your 'Documents' folder) then click OK (or Save). The GPX files are very small, so it won't take long to save.
Next, you'll need to copy the GPX file onto your GPS device. You may have to read the manual for your GPS to find out how to do this, but I use Garmin devices so I'll show you how they work. Once you've plugged the GPS in using a USB cable, it will appear as an external drive on your computer. You need to copy the GPX file(s) from where you downloaded them to (your 'Documents' folder) to a folder on the Garmin external drive - on some devices it's a folder called 'New Files', on older devices it is called 'GPX' ... refer to your device's manual if you're not sure. Once copied, you can safely remove the device from your PC and then disconnect it.
Using the GPS to navigate the route will again depend on your device, but on my older Garmin Colorado I go to Shortcuts, Where To?, Tracks, select the track, then 'Go'. It will calculate the course, then show it on the map. There are so many devices out there with different ways of doing things though, so you may have to check the manual for yours if you're not sure.
If you don't yet have a GPS device, Garmin's eTrex range are a great way to get started.