I know a thing or two about the hospitality industry so I will try my best to answer this. I do not think there is a single way to advance in this field and it depends on the type of cuisine/restaurant you want to open or work at. But let's say you want to be the head chef at a fancy modern restaurant in the city.
One option is you can go to culinary school which will teach you the basics of cooking, this is what a lot of cooks do. Another option is to find a chef or cook that you think is good and try to get work with them.
To make it to the top in the culinary world can be a bit grueling. It's long hours in a hot kitchen with hot objects and sharp knives at every corner filled with fiery personalities. I recommend taking the time to work in some kitchens as a prep cook or even a dish washer to really get to know every position in the kitchen. If you want to be a head chef you'll want to know how to run everything.
Anthony Bourdain started off in the back of the house and worked his way up. I personally think this is the best way to move up and really get to know your field. A lot of chefs will travel and seek out opportunities in the top kitchens. It's not a glamorous life style at first and often times you will not be paid that well, but I think this process builds a strong foundation for anyone pursuing a career in the culinary arts.
Be sure you really have a passion for cooking, do you already cook a lot at home? Are you naturally hospitable and a good host to your family and friends?
Aaron recommends the following next steps:
You will run the back of the house- keeping energy high, your teammates engaged and motivated, and managing day to day personnel problems (e.g., when the potwasher doesn't show up and you have to find a sub).
Enjoy the career, knowing it is long hours, hard work, but a rewarding and fun job.