It is a good idea to mix the compost regularly as it is supposed to be an aerobic process (with oxygen). For this we have this great tool, like a big bottle opener. It is then a matter of turning and pulling, to get humidity, different ingredients of the compost, the air pocket well disseminated in the compost.
the other solution we have is our cascade compost system: when the top is full it is pushed into the below part, and get turn over in the process. It is important to always keep the right humidity, not too much, not too little :)
What do we put in the compost? everything that composts! yes even orange peels or garlic. What is important is to add in the kitchen compost a fair amount of grass, papers as sources of carbon, as well as making nice air pockets. In Can Decreix we also need to add regularly some water as the climate is very dry.