DoTTGaMMa has it on the dot. Just remember that you have to resize the logo for each different use of the logo. When you initially create it, create it large - you can then make it smaller. When creating logos view it large and small - if you're losing details, simplify your logo.
**Thought: You really should just do a logo design in vector... it can later be resized and you don't have to worry about all the other stuff I say**
My best suggestion - start with your logo in RGB, Set the document size to 5" x 5", and your resolution to 300dpi. Unless you're planning on REALLY going crazy and making the logo the size of a building, this should cover pretty well any uses you can come down to later (if you're including HD video... not sure, but you may want to go bigger).
Website Resolution : 72dpi is typically the standard.
Print Resolution : 150dpi is good to go by for most printers, but 300dpi is best (printers will ignore the information they can't process anyhow).
Hope this helps!
*Edit:
Note: DPI doesn't mean anything if you're going to set your pixel value anyways - and never comes into application on the web. If you're using DPI it's used to specify how many "Dots Per Inch" there are when you go to print your document.
Largest application for a logo that I can anticipate would be a billboard. Since I don't know final output sizes I'll just use this as an example. If you're printing to an 11"x17" sheet you're likely to be seeing it from up close and will want 150dpi to 300dpi, and would set your final document size to be the size of the output. If you're printing to a poster, and don't want to die waiting for your computer to load you could set your document size to be smaller (posters are seen from farther away, same with billboards) and could get away with setting a higher dpi and stretching the image to fit the poster/billboard later.