So it all really depends on what kind of graphic design you want to be working with. Are you doing print, web, or something else? Are you working in vectors? If so you'll be using illustrator or are you focusing raster images, where you'll be using photoshop.
Although I do use illustrator occasionally, I focus mainly on photoshop and I've found that
this website has helped me alot. Some of the tutorials are paid but the free stuff is great too. The way that I learned photoshop was to google "photoshop signature tutorials" and just follow those and I eventually picked up a working knowledge of the program.
As for general tips I would say that whenever you're doing something, try and use non destructive editing so you leave the original source of whtever you're doing intact. So using layer masks instead of erasing part of an image, or importing render into a canvas as a smart image so you're not losing quality if you're sizing up/down.
I'm always more than happy to talk about design as it's a great passion of mine so hit me up for more info or anything