If you just want to design logo you can use adobe illustration or if you want to some than a logo I personally suggest you use Adobe Photoshop its really good for designing and photography. I use to make Logos From Zega Logos A complete service for designing card, flyers, business card, logos and website
I think free logo maker can be good option for you. I prefer to check this one https://www.logaster.com/logo/ I often create different logo for my online shop or blog. Such kinds of logo makers are easy and simple way to create logo. You need to follow the instructions and make few steps for creating needed logo.
Photoshop and Illustrator are the most convenient tools, you can do pretty much everything with them. IIIllustrator is vector based, so no matter the measurements its quality is going to stay the same. Photoshop has so many tools you are pretty much limitless with it.