I believe that with the arrival of things like SVG, CSS gradients and RWD, Photoshop is becoming less relevant in the world of web design. Gone are the days of creating those 1px background gradient maps and designing a pixel-perfect page layout, then slicing it up into web-optimised image files.
For me, the clue is in the title. Photoshop is best suited to working with photographs and the like, which is mainly what I use it for.
That’s not to say it’s exclusively for that. There may be some web graphics where it suits. Like for textures and patterns, particularly if you are going for a more natural, organic or grungy look.
But for more the common, crisp, clean graphical styles, vector graphics are the way to go. SVGs are perfect for things like logos and icons on the web, being resolution independent and super lightweight. When in-lined they can take on some level of interactivity, being assigned css styling to alter properties which can allow things like hover effects for example.
I have pretty much weeded out all PNG graphics for logos and icons in favour of SVG.
So for logos, you really should be looking at vector graphic software like Illustrator, Sketch or whatever.
And of course keep Photoshop to hand for re-touching and optimising your photos.
As with most things, you gradually become less terrible at it. 