You don't need a piece of acrylic, but cardboard won't do the job - you can use an appropriately-sized scrap of wood or a piece of particleboard.
When you buy a router, it probably won't come with any bits. For routing cavities, most people use a "flush-cut" but, which has a bearing the same diameter as the bit. The bearing runs along the template, producing a cavity the exact size and shape of the template.
Here's an example of such a bit:
http://www.leevalley.com/wood/page.a...168,46171&ap=1
Hope that helps.