Quarter inch is probably right, but you need to check the router as it will tell you what size shanks it accepts.
Cheap router bits work great; they just don't last as long.
If you look at the pic of the router bit in your link you'll see a wheel on the tip of it. That turns on ball bearings and runs up against the edge of the piece you are working on. You set less than half the base of the router onto the piece you are working on (so the bit extends down past the edge of the piece), turn on the router, then move the router into the piece. That wheel will hit the edge of the piece and keeps it from moving past the edge. You then slowly push the router along the edge, keeping it flat on the piece and with enough pressure on the side to make sure the wheel maintains contact with the side of the wood. The direction you move is so that while the bit is turning it 'bites' into the wood, rather than the bit pulling the router along as it turns. With a bird's eye view, if the router bit is turning clockwise, then you would move the router around your piece in a counter-clockwise direction.
You need to be sure to set the depth of the bit in the router correctly. If it is not perfectly set up you will not get the profile cut you are looking for. So make sure you set it then test it on a scrap piece of wood to get it right. I don't think I've ever got it right on the first try.
Hope you can picture what I've said above. I'm sure Youtube has many videos.