Before you do anything, call DIG SAFE or its local equivalent. Hitting a gas line will ruin your day. (Hitting a sewer pipe won't be much fun either.)
You could rent a tractor with a front-end loader, a wheeled loader, a skid-steer with a loader ("bobcat"), backhoe with a front-end loader.
If you don't need to haul the soil away, you could rent a small bulldozer. With this strategy, you'll need a place to push the dirt.
If you are hauling it away, you'll want to think about the truck and how you'll unload it. In other words: shoveling dirt out of the back of a pickup is a hassle, consider a dump trailer, see below.
Regarding skill level:
- At the low end, wheeled loaders (tractor, backhoe) are probably the easiest to use. (I've been driving tractors since I was a kid, so I'm guessing at the skill level required for a novice.)
- Be careful when dumping into the truck (or trailer, see below) not to hit the sides with the bucket!
- Be careful when dumping into the truck that you don't bump the front of the machine into the side of the truck!
- Be careful of overhead clearance near power lines!
- Some machines have the tendency to tip forward when the bucket is loaded. If you don't have a lot of weight in the back, beware of tipping forward. (Especially when you're next to the truck...)
- Beware that the arm hanging off the rear of the backhoe is easy to bump into stuff while you're backing and turning. I'd avoid the backhoe if you have any kind of obstacles and there's any other machine available. Otherwise be very aware of where the arm is while you're maneuvering around the yard.
- Getting the soil even and level as you work is harder than it looks, and takes a fair amount of skill and practice to do quickly. I own a tractor with a loader, I don't use it much for this kind of work, and it takes me a while to get an area somewhat level -- and then I end up finishing with a hand rake. I know current/former full-time equipment operators who can do the whole job with the tractor in about a quarter of the time it takes me, not including the time I'd spend raking. (And I've been doing this a while and get semi-regular practice -- the point is that it takes a lot of practice to get good.)
- The controls on the bulldozer might take some getting used to. I haven't used a bulldozer but I know enough that the controls operate differently.
Things to keep in mind:
- As Scott Bruns mentions in his answer, make sure the loader can reach the truck you'll be loading.
- You need to move about 25-35 cubic yards of material. A standard full-size pickup truck can hold about 2-2.5 yards. Beware that a yard of topsoil weighs 2000-2500 pounds. So a half-ton pickup can only carry about half a yard. That's 20 loads or more. Consider renting a dump trailer. Just make sure your truck has the tow capacity for the amount you load on the trailer. A trailer will generally reduce the clearance/height requirement for your loader; you won't have to lift as high. And a dump trailer reduces your labor requirement for unloading. (The trailer rental fee is much cheaper than the cost of repairing springs on your truck.)
- You're going to remove the top 4-6" of soil. What's underneath that? If you have really deep topsoil (12" or more), then you'll maybe do ok. But a lot of residential developments have maybe 2" of topsoil over the top of cheap fill (which is probably either heavy clay or very sandy depending on where you live and what's available nearby).
- If you're in the shallow topsoil situation, you're going to have a hard time getting a lawn to grow where you remove the topsoil.
- You may need to remove extra material and then add back good topsoil.
- You might be able to get away with eyeballing the area to get it level. But if you want to do a professional quality job, get a surveyor's transit and use it to level the area. (You'll need a helper unless you get a laser.)