The Oatly oat milk brand has been a game changer for me. It provides the texture of milk without adding a noticeable dairy (or soy or nutty or coconut) flavor.
Experimentation will reward you! There is no right way to tea, only some right ways for you.
I'd like to try Oatly, but it is not sold here. I tried making oat milk myself, but it turned out terrible. I actually quite like the idea of vegetable milk and have tried making a bunch myself: soy, oat, rice, coconut and peanut. Of those the best ones are coconut and peanut. Unfortunately cocunut has a strong characterist flavour, so it doesn't work everywhere, but peanut milk is really very good, and has no noticeable taste, it's good even to drink plain, I recommend you try.