When sautéing vegetables, should I wait for the oil to heat up first?

In addition to the fact that some oils get quite a bit hotter than other, (so that hot with one oil isn't the same as hot with another), there is an in-between here that some cooks rank as a default option whenever inexperience prevails, namely warm oil. Regardless of which oil one is using, this can be achieved by applying an initial lower temperature to the sauté pan and then cranking up the heat once the vegetable(s) are in the pan and can be visually assessed for how they're performing.

Hot oil with vegetables high in water content risks splattering or worse (fire) when suddenly introduced. It is also easy to burn more delicate ingredients such as garlic (burn, in this case, meaning no more than browning). Meanwhile, ingredients porous in nature (such as mushrooms) added to cool oil are certain to absorb (and retain) more of it than you would perhaps hope for. Stated here, (The Reluctant Gourmet),

There really is no ideal temperature. It really depends on what you are sautéing and what you plan to do with the ingredients when you are done.

Since it's not therefore difficult to see that a rule of thumb toward how hot cannot be established, (though no saucier recommends cool, nor the wiki on Sautéing), decisions must be based on knowledge which is specific to the ingredient(s). If uncertain, go with in-between.

Cooking with oil is a step by step process since you have to take care of few things so that the balance in food is retained. Being an Indian, where we use different kinds of oil in our daily foods, let me tell you few things which you need to remember while handling oils:

• Firstly, put the pan on fire and let it dry completely in case it was just washed so that the water droplets go off completely otherwise the oil will spill out and create a mess. In general, heating the pan first is always a good thing to do for various reasons.
• Then pour in the oil and let it get warm. This enables the oil to become 'thin' and the oil becomes easy to digest. Also warm oil is good for the vegetables going in it because they get the correct temperature and helps them cook faster.
• Now all you need to do is add your vegetables preferably with a little bit of tempering which enhances the taste and flavor. You could then stir fry, saute or do whatever you want to do with the oil!!

All the best!!

You might check out this thread which spends some time addressing the oil-then-heat vs. heat-then-oil dilemma:

Do you heat the pan first, then add oil? Or put the oil in and heat up with the pan?

Also, for the sautéing of vegetables, the oil should be hot before you add the vegetables to sear them and to induce the Maillard browning reaction. Heating the vegetables along with the oil can give you oily vegetables (the sear will prevent that).

For what it's worth, the word sauté comes from the French verb "sauter" which means "to jump" - it describes the way vegetables jump when introduced to the hot oil of the pan.