The liquids suggested so far are either flammable solvents, or CFC refrigerants (bad for the ozone layer and tightly controlled.)
I can tell you from experience that a measuring cylinder of butane boils with an appearance like water but is pretty cold (around 0 C) so I dont recommend putting your hand in it (liquid nitrogen actually feels less cold.) Whatever you do, with flammable solvents, do it outside and away from flames!
Solid carbon dioxide "dry ice" in water is a better option. there are many videos on Youtube, for example this one which illustrates the effect. It also shows one of the problems: the fog goes down, rather than up, due to CO2 being denser than air. If you want the fog to rise this will also be a problem with other materials. A fun thing is to add some indicator to the water, whose colour is affected by the acidity of the CO2 dissolved in the water.
The fog is largely water condensed by the CO2 (as the vapour is produced at the sublimation temperature of CO2, -78C.) If you want vigorous effevescence without so much fog, you could look into the reaction of acids with carbonates and bicarbonates. If you mix solutions, you will find the reaction instantaneous and the material will bubble so fiercely it jumps out of the pot. Solid carbonates react much more slowly, and you can fine tune it with both the acid strength and the particle size. Here is an example.
EDIT: Another reaction that produces a harmless gas is the catalytic decomposition of hydrogen peroxide solution to water and oxygen. You can fine tune the speed of reaction very easily by varying the amount of catalyst. you can see this both with and without the optional dishwashing liquid here (turn the music down if you don't like it!) and there are far more dramatic demonstrations on Youtube. It should be borne in mind that the more dramatic demonstrations this do generate a lot of genuine heat though.
If you want smoke, commercial smoke machines often use propylene glycol (which has a higher boiling point than water) and is much less toxic than ethylene glycol (as it is metabolized to lactic rather than oxalic acid.)