I try to be diligent about brushing my at least hair once a day (I use a boar's bristle brush because it moves grease from the roots down through the ends. Vegan comrades might be able to suggest an alternative that works the similarly but I don't know of one off-hand). I use hot water on my scalp when I get in the shower and kind of give yourself a scalp massage to help move some of the grease down your hair. I use a cup of water with a couple of teaspoons of apple cider vinegar as a shampoo replacement and leave that on for a little bit. I co-wash (so I use conditioner afterwards) mostly because it smells nice but I don't think this step is necessary. Rinsing with cold water is unpleasant but helps close the cuticles and makes your hair super shiny. My hair is thick as shit and required a lot of shampoo when I was using it and was still always dry. Not washing my hair with shampoo saves me a lot of money and my hair basically doesn't get greasy anymore. It doesn't work for everyone though and it's definitely uncomfortable and gross at first lol

Edit -- just read over this and it looks kind of impenetrable lol. TLDR is brush your hair, use hot water when you massage your scalp in the shower.