Take a chainsaw to your brushy fence line and clear it if you want to go with electric. Goats can be hard to contain with electric though, you gotta make sure they respect the hot wire.
We've found that field fencing is the cheapest wire fence per foot; it's around $175 for 150 feet last I checked. This is the fencing that has 6x6" holes tapering down to 2x6" holes near the bottom. It's heavy gauge and usually 4' tall. In some ways it's way more durable than welded wire, and in some ways it's not. Welded wire will start coming apart if a goat or other animal works it too much. My goats have mastered tearing welded wire apart. So woven wire, like the field fencing, is the solution for them. Except the field fencing has such big holes that it's not vertically sturdy. It's easily crushed and warped if it's not fixed tightly to posts.
The solution we use for both goat and pigs is field fencing with 2 strands of hot wire. One down low and one at chest height for the goats. The field fence then doesn't have to endure abuse from the animals, but you're not relying solely on a hot wire to contain everyone. You can even add a hot line on the exterior of the fence to fry curious predators!
As mentioned above, you could also create 50 or 100' run lines through the forest; they're pretty cheap at a hardware store. Lash the line between two
trees and have one goat per line. Clip their halter to the runline's pulley with a 3' lead (that they can't chew through) and the goat can now run that length of the runline, and also a few feet on either side of it. I like to do this with my goats in unfenced areas that need the brush taken back. Then in the evening I can put them back in their secure goat
yard. This method is portable and doesn't hurt the trees you hook the runlines to.
You gotta make sure that their lead is short so it can't get tangled up in the brush. And only 1 goat per run line! They will readily hog tie themselves together if you run them too close to one another.
Edit on the run line idea: We've used conventional dog run line wires with pulleys, as well as dog tie out leads. Tie out leads are cheap
online, like ebay. You can pick up a 50' 120lb rated lead for under $20. Sometimes what I do is clip one end of one lead around a tree about 5-6' up, then on the open end, clip it to a 10' lead. Then the open end of the 10' lead gets clipped to another 50' lead, and the open end of that lead to another tree. So what I have is 2 lengths of 50'~ run line, and a 10' barrier space between them. This allows me to put the goats on the same path and same setup, but it keeps them far
enough apart that they don't get tangled, if that makes sense. Short 3-5' leads are hard to find but can be made for less than $5 at the hardware store. Just get some rubber coated security cable by the foot, some wire clasps and some fastener clips, like the lobster clips. Mount a clip to either end of the 3' cable and pinch the wire clasps closed. Boof! A short cable lead! You clasp the lead to the coated dog tie out lead and the goats can zip up and down the length of the lead the same as a dog tie out lead/pulley setup. Hope that makes sense