Get high or get high on drugs

You can use either, depending on the audience. Drug users will know what 'get high' means; you don't have to add 'on drugs'. However, non-drug-using people may need to have 'on drugs' added to make the meaning clear. Incidentally you can get high by inhaling glue vapour or nitrous oxide, neither of which are 'drugs' in the conventional sense. It may be more sensible to specify the drug used, e.g. high on marijuana, amphetamine, LSD, ecstasy etc.

Or cake.

You can get high on things that have nothing to do with drugs.