Hey Man,
Take the outlet (that part that connects the gutter to the downspout) to the local hardware store and buy a hole saw the right size. Then just drill out the hole, caulk around the hole with silicone and pop in the outlet.
Make sure you re-pitch the gutter so that water flows to your new downspout! To do this, just loosen all the nails on that half of the gutter except the very last one at the downspout.. Pull that one out last, then lower it an inch or so then put the other nails back in again working towards the middle of the gutter.