I have a template and a router, and am told I need a plunge bit. If I put the template on top of the wood, how do I start the hole? Do I cut a hole in the wood to have a starting point then follow the pattern?
The bit on the left has spiral flutes and a tapered end, so it can plunge (ie, you can press the board straight down over it and it will drill into the wood just like a drill bit). The bit on the right will not, because it's square on the bottom and does not have a spiraled flute.
Here is the article that is the source of the picture, with an explanation: http://www.craftsy.com/blog/2014/12/router-mortising/
It really depends on the type of bit you are using. some bits can plunge by resting the edge of the router and letting it cut through. If there is a bearing or non cutting bottom drill a hole with a drill bit large enough to get started and have fun.
This can largely depend on your template. Are you going to be cutting through the wood making a complete cutout or are you making a recessed design? alfreema's answer covers the bits that you be using in either case.
However you are not locked to just those tools. Your basic drill can get you started. Again, depending on your template, you can use something like a jigsaw to remove more of the waste. This has the advantage of clear more of the wood and putting less stress on the flute bits. Which are generally more expensive that a jigsaw blade.
Don't forget to clamp the template down as you will be putting pressure on it while you are moving around with the router.