Michelle, how big will it be? Do you have a design in mind? Will it have a top, how tall will it be? It is easier to make suggestions if I know what you are dealing with.
I think you are asking specifically about wire mesh. Different wires give you different levels of protection. Some of that will depend on the gauge and mesh size for the type of wire. The heavier the gauge the harder it is to tear. The smaller the openings the more critters they keep out and the more protection they offer. Often the weak spot is not the wire itself but how it is connected. I don't know your risk tolerance either. How much would it devastate you if you lost one to a predator?
Most chicken wire is pretty light gauge and might have holes 1" or 1-1/2". It will keep some critters out but medium to large dogs, raccoons, or coyotes, maybe possums and some other critters have been know to tear it. Some people use it for decades without problems but others find themselves wiped out pretty quickly. It provides some protection but not that much.
Hardware cloth is better but is more expensive. It also comes in different gauges and mesh sizes. Really large dogs or big boar raccoons have been known to tear it but it is a pretty safe choice. It will stop most critters.
I use 2" x 4" welded wire. I can't remember the gauge but it will stop just about everything up to a bear. One issue though is that the holes are pretty large. Certain smaller critters like snakes, rats, and many weasels con go right through. There are other gauges and mesh sizes that might work better for you.
If you are going to cover it, some critters like raccoons can climb up to the top and tear the lighter materials. If all you are worried about at the top is birds of prey, chicken wire or even deer netting can work.
One way to greatly improve the efficiency of any of the fences is to use electricity. For a larger run it may be more cost effective to use a cheaper wire and electricity.
I don't have just one answer for you, there are too many variables. Chicken wire will keep chickens in but isn't all that good at keeping other things out. The others can get pretty expensive pretty quickly but do a better job.
Good luck!
