The best and most fun way to quickly level up is by simply playing a lot!
Actions like serving customers, harvesting crops in the Backyard etc. will give you Experience Points (abbreviated as XP)! The more XP you have, the more Appliances, Crops and items you will have at your disposal - making you a better chef as you progress in the game!
So, keep serving customers, harvesting those crops, producing those goods and sending those orders.
Chef! Within no time, you’ll be the greatest Star Chef ever!