How can I differentiate between organic and inorganic compounds in the laboratory? By flame or solubility in water? What should I do first?

There is no 100% definition to differentiate between organic and inorganic compounds.  There are some general ways to tell the difference between them.  Organic compounds tend to be composed of the elements carbon and hydrogen.  Inorganic compounds are less carbon, more oxygen and hydrogen, and incorporate a lot of the metallic elements.  Another difference would be that organic compounds are the building blocks of living things, while inorganic compounds may or may not be involved with living organisms.

To answer your question directly, there is an old saying, "oil and water don't mix."  Oil is an organic compound, composed of the carbon and hydrogen atoms mentioned in the first paragraph.  Water, while it is necessary to life as we know it on earth, has an inorganic composition, H2O, and will not mix with organic compounds.  I would vote first for the solubility test, to see if the compound will mix with water.  If it will, it is probably inorganic; if not, it is organic.  For flammability, organic compounds will combine with oxygen as they burn, while inorganic compounds, depending on the chemical composition, may or may not burn.