I tied my goats up at night, either to the vehicle or to a convenient tree (watching to see that they didn't do too much damage). Since I use horse lead ropes, which are fat and soft, the ropes don't hurt the trees (might with a thousand pound horse on the other end, but the goats don't pull that much). Actually, they were usually tied during the day, too, unless we were walking them.
A picket line would also work, probably better, but I'm not set up for one yet.
Most parks, state or national, won't allow goats. As far as I know, all national forests do as long as they are kept leashed (and some places you don't have to have them on a leash, but those would mostly be places that don't see a lot of traffic). You would need to check with the ranger station closest to the campground in order to find out what a specific campground allowed, because they aren't all the same. The same goes for BLM land and campgrounds -- find their office and talk to them.
And if you want more information on packing with your goats (even a dairy doe can carry a light load for day hikes), check out the North American Packgoat Association on-line.
http://www.napga.org/ Don't be deterred by the cost of buying equipment, because the cross-buck pack saddles are pretty easy to make, and cheap daypacks will work for panniers, at least to get started with. I use thick wool blankets that I found at the thrift shop for saddle pads (my best one looks like it was run through the washer and dryer and shrunk, but is very thick) -- I cut them to size, and blanket stitch the edges if it starts to fray.
Kathleen
Kathleen