Oil paints, acrylic paints, watercolours and gouache are all non hazardous for flying. Only some oil solvents have a flash-point that is below the temperature allowed.
Most airports do not allow liquids or palette knives through security so you will need to put your paints in your checked baggage and not carry them on with you. Always check beforehand with your individual airline. Of course you may be unlucky and security staff may become alarmed by the idea of paints, wrongly thinking they are flammable, so taking Material Safety Data Sheets (MSDS) and being sure of your information may prevent them from disposing of your paints.
If you wish to use solvents, some are non-hazardous for flying, though again, the security staff may not be convinced. You may wish to purchase your solvent at your destination, which would also avoid any chance of spills in your luggage.