I started last year. Ferment completely dry either a Chardonnay, Chenin Blanc or a Pinot Noir Blanc (3 most common). Your acid levels have to be right as well as all the sugar eaten up. Age to ensure all sediment is out.
Restart fermentation with a specific amount of sugar and a different yeast. Once fermentation is underway bottle in champagne bottles and crown cap. Gently shake bottles once per week for a few months to redistribute yeast and sugar. For a year age in the bottle shaking every month or so. After a year turn bottles down (riddling) to get all the sediment to the neck.
Once all sediment is in the neck, chill, then freeze the neck. Pop off the cap and the CO2 built up in the bottle will blow off the sediment plug. Immediately place your thumb on top of bottle to stop loss of champagne. Top off with similar wine and plastic plug and wire ties. Age and drink.
Basically that is the process. If you are more serious I can get a few more details.