What milk actually does
Milk softens the drying grip of a strong tea and rounds its edges, and it adds fat and body of its own. It also muffles aromatics. That last part is the whole issue: in a tea whose interest lies in delicate aroma, milk removes precisely the thing worth drinking it for.
Teas that take it well
Strong, brisk, high-tannin black teas — Assam, robust Ceylon, English and Irish breakfast blends. These are frequently blended and graded with milk in mind, which is why they can taste harsh and one-dimensional without it. Masala chai is built around milk from the start and does not work any other way.
Teas that do not
Green tea, white tea, most oolongs, and any black tea sold on the strength of its aroma — Darjeeling, Yunnan golden bud, delicate Chinese blacks. Milk flattens them. This is not snobbery: you can put milk in whatever you like, but you will be paying for aromatics you then cover up.
Milk first or tea first
A genuine historical argument with a mundane answer. Milk first protects fine porcelain from thermal shock and mixes more evenly; tea first lets you judge the colour and adjust. There is a plausible flavour argument that adding hot tea to cold milk scalds the proteins less, and it is a small effect. Do whichever you prefer and expect someone to disagree.