It depends a bit what you are going to do with the mince. I have got no opinion on lamb because I don't eat it.
For turkey I prefer the dark meat to the white because it has more flavour, although breast mince makes a decent meatloaf. The dark meat has more fat than the white, the white can be a little dry when minced. If you want to make a turkey breast burger, then you need to add some moisture and some flavour, also double or triple flour and egg them to stop them falling apart.
For beef again, it depends what you are going to do with it. I'd probably buy decent quality ready minced beef tbh and for a luxury home made burger, i would rather chop the mince with a knife than mince it.