OK, lots to cover there... Do a search for cattle panels and select the cheapest then see if they have a store near you. Down here TSC has them on sale about once a quarter for $19 and change each. Otherwise they are in the $21-23 range. Not a substantial difference when you're only purchasing 4 or 5 panels... $15? really, not a big deal. If you were purchasing 100, then yeah, a buck makes a diff.
Another option you can check into that many use is electric net fencing. Many here have the one from Premier one
https://www.premier1supplies.com/ Of course there are other brands as well that are a bit "cheaper" (less $$ and less quality). But it is a pricey option... Probably much, much more than you wish to spend, but in fact would be the BEST choice for in the woods. It comes in I believe 160' lengths and it is light (roll it up) and easy to move. Step in stakes hold it in place, so no need to damage trees with nails or staples with regular fencing.
You'll need to move it frequently. Goats will clear that sized area in less than a week, and I mean from the ground up, to as high as they can reach standing on their hind legs. They will chew the bark off young trees, effectively girdling them and killing them.
Deer present several issues... Some here don't like the deer around because they carry parasites that can infect your goats. I DO LIKE the deer around because I enjoy watching them and when in season & properly seasoned, they are very tasty.

Having said that, I haven't killed a deer in many years and may never again. Nothing sentimental, just don't need the added work of processing it. Deer carry meningeal worms which can be deadly and hard to get rid of. IMHO, it's kinda senseless to fret about it since the deer have been all over your property and if they are there, they're there.

Can't kill off all the deer to protect a few goats. Nor can you "sanitize" your entire property or where the goats will be to kill any/all parasites. Just not realistic. You deal with issues as they worm their way into your life. Since a deer calls an area of about a square mile home, they won't strip/destroy the browse on your property. The goats will because they are confined to just that area, hence the need to move that woods enclosure about every week... give the forest time to re-grow for the next time they feed there.
Goats aren't "intentionally" destructive... What they're doing isn't destructive to them... That's a term we assign to what they do. They are hairy and live outdoors. They get itchy. When you get an itch, what do you do? Scratch it of course, and so do they... on the fencing, by leaning into it and walking, which stretches it and will eventually break the welds on welded wire. When the cookies you want are on the top shelf of the cupboard, what do you do? You reach up to get them and if you still can't reach, you get a step stool or chair to climb on. When the goat (or some sheep) want those tasty leaves above their heads, how are they gonna get to them? well, why not climb up that fence right there to make reaching them easier? Or... HEY, what's THAT I see way over THERE -----------> Maybe I can get a clearer view by climbing up this fence here... Which will break the welds on the welded wire.
As for predators, nothing smaller than a coyote would normally be an issue. I say normally because when a "pack" of little dogs gets together, they are just the same as a pack of big dogs and will take on the same target animals/prey. A group of small dogs can and will kill a goat. Also, a fox or smaller predator will go after and kill/eat kids (baby goats). A coyote (dog, deer, etc) can jump right over a 4' fence, including electric. They (anything/one) only get zapped if they are in contact with the ground and fence simultaneously. That's why birds can sit on a live electric fence and NOT get zapped. Need to be in contact with the ground to complete the circuit. The fencing is primarily a deterrent... It won't stop a determined killer from getting to your goats. Goats do jump, but most walk up to and press against things long before considering jumping over them. Once they do this and get shocked, they will generally not challenge the electric fence a second time.
Having a lockable shed/barn (with plenty of proper ventilation) to keep them in at night is ideal for safety and shelter. Not everyone does that but of course they should have SOME sort of shelter from the elements, especially when it's cold and wet. If I recall, you live in PA or MA, or someplace where you get cold and snow.
I think I've cover most of it...