You might not know it when you look at me but my color is enhanced.
Sulfate-free shampoos are best for color-treated hair.
I'm going to have to say that I don't know a ton about hair products, but I'll try to share what I do know. The best sulfate-free shampoo that I've found for my oily scalp is L'oreal Oleotherapy (in the sunshine colored tubes!). As far as this goes, I've been using a couple of deep scalp cleansing shampoos and they haven't been stripping my hair color too much.
Since this is your first time (I'm assuming?) with hair dye, it's important to note that you'll definitely need to condition your hair regularly now. Only use conditioner on the hair that's not touching your scalp. Like, just the mid-shaft and ends.
As for your scalp in general being oily, you might want to look into a scalp scaling treatment like this Lador one. Product buildup from shampoo, conditioner and any other things you might use can clog up the follicle and also leave a kind of waxy coating on the hair itself, which makes your hair look and feel greasy. Scalp scalers and buildup-removing shampoos can help eliminate that problem.
They're not great for colored hair, but I do them occasionally anyways because washing my hair every day is much worse for my colored hair.
Also make sure you're not washing your hair with super hot water for the same reasons ABers recommend not washing your face with hot water. Remember, your scalp is basically just an extension of your face covered with a mane.
If you need something on greasy roots, use a little dry shampoo or baby powder to mop up the oil.
Once your scalp gets used to not being scrubbed every day that should slow down the oil production a bit. Also, spearmint and evening primrose supplements are good for managing oil on the face, so maybe worth a shot for the scalp?
I always rinse the conditioner out last to give it a little more time to do its magical conditioning goodness.