I like my arena well fenced. If I fall off, I don't want the horse to run off and hurt himself or someone else or someone's child, and that's a sufficient reason to insist on it.
It's good to have at least one solid escape proof enclosure to put your horse in in an emergency, or to put a new horse in before you turn it out with the others, and the arena is perfect for that.
Further, you can't free lunge in unfenced areas; it's harder to train shoulder in/out or sidepass or even turn on the haunches without a good fence.
If it's just a few of you and you're doing a lot of groundwork, I wouldn't worry about the footing. Grass is good. As long as it's free of potholes and rocks and stuff that could turn ankles and hurt hooves, it'll be fine for your purposes.
Fences must be good. I like wood; it's resilient if your horse runs into it and bends a little, which can really help prevent injuries. The plastic boards that look like wood are nice, but pricy and can be very brittle when they're cold. I know wood is pricy, but if you can contact a lumber yard and order rough cut (just means it hasn't been planed smooth) oak boards you'll have a fence that will last for many many years. You can also bargain a bit - many lumber yards will have piles of "culls" in the back that they'll sell, sometimes for just pennies on the dollar. A little bend or deviation in the boards will make them unsellable for the home trade, but just fine for a fence. Same with posts: a 4x6 treated post for a pole barn has to be straight; if it's warped a little they'll often discount it for you for fence posts. I've also used landscape timbers, but they don't last as long and they tend to be a little flimsy.
You can rent a 2 person powered post hole drill to help set the posts yourself.
Size... make it as big as you can. I like the size of a small dressage ring (20x40 meters) because then I can practice dressage more easily. That'll hold several jumps. The minimum would be what you see in indoor arenas. You could probably shrink it to 50x50 feet and still get some good use out of it.
It's a lot of hard work, but it's well worth it! Good Luck!
Edit: I recommend against high tensile wire, or wire of any sort. It's only safe for horse fencing if it's electrified, and it can't be electrified if you're working in a ring made of it. You can't have him spooking madly if he (or you) inadvertently touch it. If you fall on it, it can really slice into you, or into the horse if he gets a leg through it. I've had two bad injuries from horses getting legs through high tensile in pastures, and no longer use it between fields. One of the horses severed two tendons and required surgery. (Power outage; the electric was off :( )