If your hair gets greasy easily, it won't dry out from washing it often. My friend has hair that gets oily throughout the day, so she washes it daily and it's fine.
Like others have suggested, there's shampoo for color treated hair, and it's more gentle than regular shampoo. I'm not sure what kind of hair you have though, it also depends on that (it looks medium thickness, and wavy?)
I think your hair is greasy because it can be 2 reasons: First. Depending on your age, your hair is at turning point. It hasn't found its origin yet. Hair needs time to complete to have its final characteristics. Second from my experience, you are washing your hair more than you need. So hair becomes greasy, because it gets used to that routine. Don't wash it everyday. Try to wash two- three times in a week. Yeah till it will seem like dropping oil. It will get used to be dry. I had the same problem. Now I am washing three times in a week. Problem solved. Also don't use conditioner more than green bean size because they also make your hair greasy. Try to apply conditioner to end point of your hair, not to origin of the hair on your head.
Um with your issue with greasy hair hun.. Use shampoo and conditioner (for colour treated hair) wash it every second day to prevent your hair from getting really greasy. It will help (its like not washing your body, if you dont use body wash to wash your body your going to feel icky same goes with hair) so just make sure you wash it with shampoo and conditioner! (you definitely need conditioner with the shampoo) hairstyles? Im not quite sure but other girls will know that 😏