Cutting a Rod
Work with a variety of materials to get a feel for how aluminum, brass, stainless steel and plastic cut. They cut completely differently and require different tools and techniques to obtain perfect results. Although it may seem like a waste of time, try machining a block of brass to exactly 1.000-inch with 0.002-inch tolerance measured in any direction. Take a piece of aluminum rod and cut it down to 0.750 inch, with a step down to 0.375-inch, and a 0.062-inch radius transition. In other words; practice making it smaller and smaller. Get comfortable with your machines and tools and what they're capable of doing. Then move on to beginner projects to improve your skill set.
Fabricate an Essential Machinist's Tool
One tool essential to a machinist is a small brass tapping hammer to fit things into final position. Rather than buy one, make your own out of a piece of brass stock and hickory or ash for the handle. This gives you practice machining and milling brass, and tests your skills with wood to make the handle. (Woodworking is a completely different skillset for the lathe, so it may be best to buy a handle from a local hardware store if you're not tooled up for wood.)
Billet Aluminum Accessories
Fabricate a custom billet aluminum oil cap, breather, or dipstick cover for one of your pieces of motorized equipment. If you're still learning, maybe an aluminum oil dipstick for your lawnmower is the place to start. Don't be intimidated by the word "billet." Although it sounds high-tech, it's nothing more than a piece made from a solid piece of aluminum. You can use the existing piece as the template. Put a high polish on the piece when it's done, and take your time. You don't get a second chance at a solid piece of metal.
Nut on a Threaded Rod Puzzle
This is everyone's favorite beginner project because, not only does it allow the machinist to perfect his skills with threading, joining and finishing, but the result when done properly is a puzzle your friends will struggle to solve. The finished piece looks like a piece of continuous rod with a nut run onto it, but both ends of the threaded rod are machined to be larger than the diameter of the threaded rod. The actual design steps to make this puzzle aren't hard, but they're extensive.