Painting kitchen cabinets is, like any painting job, a simple task. But mastering the perfect glassy finish is all in the prep work. Before brush ever hits wood, there has to be a lot of time devoted to getting the surface ready to accept paint. That means properly cleaning, sanding, and priming every inch of the surface, or the finish color won't stick well.
Cleaning is the most important step in the process. Years of greasy fingerprints and cooking splatters can leave a layer of grime that inhibits paint adhesion. You can remove most of the gunk with TSP substitute (a cleaner from DAP or Savogran) or a degreaser—the former if the cabinets are not too dirty, or the stronger degreaser if the grime is thick—but it may take a couple of passes. After that, you'll need to rough up the surface with some 100-grit sandpaper to help the paint stick.
The primer you use can also make or break the finish. To get a glassy surface, you need to use a "high build" sandable primer, such as Eurolux from Fine Paints of Europe, to best fill the wood and even the surface. The sandable part of that equation is imperative, so that you can smooth the surface before painting on the finish coat. You may even need two coats of primer to completely fill the grain.
To keep the doors and drawers flat as the paint levels, make yourself a pronged drying rack by drilling screws up through several pieces of scrap wood. That way you can flip your work as soon as it's dry to the touch. Also, screw cup hooks into the edges of doors and drawers so you can grab hold and move them without fingerprinting the paint; then hang them up for out-of-the-way drying.
The formula of finish paint you use contributes to the smooth look. Traditionally, painting cabinets for a high-traffic area such as a kitchen required using oil-based paints. However, working with oils can be messy, and the fumes are toxic. Fortunately, while latex paints will never quite self-level and flow as well as oils, they're getting close. Latex formulas specified for cabinetry—labeled "100% acrylic"—will create an even, durable finish. And, in many cases, they're also low in volatile organic compounds, or VOCs, which make that noxious paint smell.
As long as you're sprucing things up, consider changing the hardware or putting on a faux finish for that added wow factor.