If you are really designing your logo yourself I would suggest that you use Adobe Illustrator as a software for this designing. Using Adobe Illustrator you will get Vector Logo which is multipurpose - can be enlarged without any resolution loss and is print ready.
If you want to use your logo only on your website and you don't have any other purpose then using Photo shop or paint is fine.
If you want to use your logo only on your website and you don't have any other purpose then using Photo shop or paint is fine.