You need to pick a size and duration for your comp. If you have multiple scenes as comps and you want to assemble them in After Effects, you can make a final comp and sequentially put all of your scene comps inside that final comp. You may prefer to render out all of your scene comps as individual quicktimes and assemble the final movie in a video editor like Final Cut Pro.
I like to make my comps 720x540, Square Pixels, 23.976 frames/second. This may seem like unusual settings, but it can easily be converted to many other formats like 320x240 Square Pixels (vimeo, youtube, etc.) and 720x480 D1 Pixels (DVD and DV tape). If I'm not sure what my final output is going to be, this is the settings that I use.
You are welcome to make your movies any size you like. Just make sure that it plays well in what ever you plan to use for output. Running a Quicktime off your desktop is totally fine.
By default, your rendered video's duration will be the duration of your work area, and by default your work area is the length of your timeline. If you don't want to render the entire length of your timeline, then you can change the work area by dragging the handles on the timeline graph (see picture above). If you hold down the shift button while dragging the handle, then the handle will "snap" to the beginning and end points of the layers in your comp. That's useful for setting the work area to the exact length of a layer in your comp.
When you are ready to make your quicktime, choose "Composition -> Make Movie..." from the Menu or hit command-m on a Mac.
Your comp will be placed into the Render Queue. By default, your movie will be rendered as "Your comp name".mov. You can click on the file name to change the name and render folder.
The default Render Settings are "Best Settings". This is probably fine for your render. The Output module is where you set the video and audio output settings. By default, the Output Module is set to "Lossless". "Lossless" will render your movie as a quicktime with the Animation Codec, which is probably too large to play well off of your internal hard drive. I strongly recommend clicking on the Output Module name and choosing a video codec that will compress your movie to a smaller size. When you are done configuring the Output Module, click the Render button to start your render.
Unfortunately, there is no perfect video codec for all situations, but here's some suggestions:
Again, if you're planning on showing your animation as a quicktime that's played off your desktop, I strongly recommend choosing "Photo-Jpeg" at 75% quality as your Quicktime codec. please DON'T render your final piece as DV-NTSC if you are planning on showing it as a quicktime, as it will look soft and pixelated when played on your computer monitor.