Getting to the first milestone at level 10 in World Chef is quite easy. For me getting to level 10 took less than one day of playing, and that was without buying any diamonds or money at all.
Basically what you do is to just follow the tasks that the master chef gives you. He always give you a notice when there is food ready to be served, costumers wanting to pay, etc.
Also, pay attention to the achievements, you can find these by tapping on the chef statue just outside your restaurant. Doing these achievements will give you both experience and diamonds. Getting the majority of your experience from these achievements will really speed up your journey to level 10 a lot. Good luck!