Here's the best priming method, hands down: 1. Place beer in keg. 2. Attach CO2 line to keg.
If you have the space and a little over 100 bucks for a keg setup, do it. You won't regret it, I promise.
As far as actual priming goes, the traditional method used by myself and many others is as follows:
1. Put your cooled (or not) priming solution into your bottling bucket. (Btw, DME, corn sugar, and table sugar all work and I can't tell a difference between them.) Cooling it to room temperature probably isn't necessary since you'll be dumping 5 gallons of 68 deg. (Fahrenheit) beer on top of it. If you're like me you'll do it anyway for irrational superstitious reasons and because that's how we did it in the 90s goddammit.
2. Add your beer to the bottling bucket using a siphon and/or hose configuration of some sort. If you want you can position the hose so that you create a neat little whirlpool that you can stare at while your beer transfers. This might help to mix the beer with the priming solution, maybe. Keep splashing to a minimum. This is actually the most important part, because now is not the time to be adding oxygen to your beer.
3. Now that the beer and the priming solution are together in the bottling bucket, you can take your spoon and gently stir. IMPORTANT: Pretend you're stirring 5 gallons of nitroglycerin that will blow you to smithereens if you create even a tiny splash. If you don't need that kind of stress in your life, don't stir it. There's a ton of anecdotal evidence which states that stirring may or may not produce more consistent carbonation, which is something you need to consider very carefully.