Warming up is simple, and it is the same shape every time: start by doing a smaller, slower version of the thing you are about to do. Easy rallies before hard ones. Shoulder circles before serves. A few short runs before you chase a drop shot. The point is to arrive at the first real point already moving, rather than asking a cold shoulder for a full swing.
Long held stretches are worth having, and we keep them for afterwards, when everything is warm and the session is over. That ordering is the part most people have backwards, ourselves included until fairly recently.
If something hurts rather than aches, none of the above applies and a physiotherapist is a better read than this page.
General guidance, not a research finding.