How should I design my logo, technically speaking?
Forget resolution and size. Any logo should be made to be infinitely rescalable. That means that it could be as small as a half inch on a telephone book entry to the size of a billboard or even bigger. That means it should be done with vector drawing tools and, that tends to mean Adobe Illustrator. The graphics done with Illustrator’s Pen tool and the text is resolution independent. You can reduce and enlarge all the graphics to any size you want without any pixelization. This is not true of graphics and text created in Photoshop or similar application. Think about this example: Suppose you follow the suggestion to make your logo 150 pixels by 150 pixels. On your monitor, the full size logo will be about two inches by two inches. But what if that logo has to be placed on the side of a truck, four feet wide and four feet tall. Each individual pixel will be over half inch wide. Talk about pixilization! Even at 300 pixels per inch, at a two inch by two inch design will still make 1/4 inch
Make sure you use 300 pixel/inch resolution when you are designing. Design the largest copy first. Size this down to get your smaller graphics. I agree that illustrator is better for this. It can be done well in Photoshop but you must size your photoshop files and maintain your text as vector layers. Once you rasterize your text by saving the files as jpg or merging layers things will get messy.
Depending on the complexity, I would use Illustrator. Photoshop is a raster based program whereas Illustrator is vector based, so when you resize it, lines and fonts remain smooth. This is because it is based on formulas rather than dots or pixels. There are some free programs that use vectors, but if you’re sending it out to a shop, most all take the postscript format or EPS file. Google free postscript or vector drawing programs. Good luck.