Do professional cooks use measuring cups and spoons?
Much more often, expert chefs will use an accurate digital scale, particularly for baking. Cups are OK for liquids, and most people use spoons for small measurments like salt or yeast, but scales are accurate for everything from flour, to honey, to water, to softened butter.

Plus, since you can generally tare scales, you don't have to wash anything out but the big bowl you're mixing everything in! Trust me on this one, a good digital scale is an awesome way to spend $16 - $25.

If you're not baking, chances are you don't need precision. You rarely need to measure ingredients, except for baking, once you're pretty competent. 

EDIT: Just a note, I bake all my own bread and I always do it by weight. I use spoons for the yeast and salt, the scale for everything else. I have a habit of using a Pyrex measuring cup to nuke cold water, I stick a thermometer in it as it cools, and start mixing when it reaches 110F. So at that point, it gets poured from the cup to the mixing bowl. That has given me many opportunities to note the accuracy of the cups (my scale is very, very accurate, I test it all the time). Even standard Pyrex measuring cups, used properly (eye level, on a level surface) will give erroneous results. For 1 cup (237ml), you can do everything right with a measuring cup and still be off by as much as 20ml either way. That's over 8%. So I recommend weighing everything but tiny spoon quantities. Sometimes, I'll even break out my gram scale to weigh salt, but I have to find the recipe pretty intimidating to go that far.