See what Paul said. Here's what I do for most of my projects, be they FIRST or otherwise.

Step one: Identify possible strategies. (ALL OF THEM!) It's important to nail every single possible method, so this stage should take a good 2 days.

Step Two: Decide on your strategy. For this, I use a Weighted Objective Table. I think there's a whitepaper on here somewhere detailing out how to use these. It's an excellent tool, so I suggest checking it out.

Step Three: Go nuts on a whiteboard. After I've found the general strategy, it's time to decide on mechanisms. Start general and work down to the specifics. First identify what needs to be done, and remember to keep it SIMPLE. If you can't do it in 2 ranges or motion or less, you should probably be re-examining step two. From here, work down to the details. Remember, less moving parts = less complexity and less failures.

Step Four: Sketch time! This is where I take the mechanism I've decided on building, and sketch it out several ways. It's good to take a highlighter to the sketch afterwards, and color code different mechanisms. This paper stage is to help me get a good grasp on what I'm building.

Step Five: CAD CAD CAD CAD! This is an extremely important step, if you'd like to be competitive. From here, I start putting my sketch into CAD one part at a time. It's important to start from the ground rules up. Start with your weight and size budgets, always keep these in mind. It's also important to design within your means while still being accurate. I suggest using the CAD for pre-fabricated mechanisms as often as possible; this will speed your design.

If you've got any more questions about CAD, feel free to PM me.