I have 6 large wooden doors with metal screen frames. What is the best approach - start with the short side and then go down the long side - do opposite sides first then do the other two sides - or something else?

THe rubber cord that locks the screen into the frame is one long piece... you don't have a choice but pick a side and work your way around the frame.

It's really difficult to get a nice tight fit. Especially when the screen is sagging at the center as you work around. Using some cardboard box pieces to raise the screen closer to level can help. They make a tool to help insert the plastic string, although you don't need it, be careful what you use, you don't want to tear or punch holes in a new screen.

That's pretty good advice... it definitely helps on the first few until you get the hang of it. You might come back to redo the first one after you did a few others.

I do a short side and then a long side. Then I put a 2x4 under each end of the screen. I deflect or bend the screen slightly in the middle by clamping the center of each long side with a clamp, this makes the screen bow a bit. Then I do the next long side and finally the short side, when you unclamp the screen it get very tight.