In my opinion, heavy gauge 2x4" mesh is fine, or good quality (not flimsy) chainlink correctly installed. One advantage of both those is that you can often find good-condition secondhand stuff if you hunt around, which helps with finances.
However I would not use either of those without something smaller-mesh on the inside of them for the lower 2-3' of the fence. To prevent chickens from sticking their heads out and predators from sticking grabby parts in. 1/2" hardwarecloth is best, but in my opinion 1/2" chickenwire (NOT the normal common 1" chickenwire) is also just about adequate, as you're only trying to prevent things being poked through and if they manage to break a mesh here or there it doesn't really matter too much.
The above arrangements will not keep out weasels or rats; and may not keep out baby possums. But you know what? Pretty much nothing will (weasels and rats anyhow), because all it takes is ONE one-inch gap and they're in.
Whatever fencing wire you buy, make sure it is GALVANIZED. Those pricing cheap hardwarecloth, make sure of that before buying. Non-galvanized will not last long...
Good luck, have fun,
Pat