From Isaac Asimov, the Complete Stories; Volume I, here are the stories I most recommend, whether you're an Asimov novice or a die-hard sci fi fan.
--"The Feeling of Power"—In a computer-dependent future, someone re-discovers how to do math on paper, leading to terrifying military consequences. Very good commentary on Einstein.
--"The Dead Past"—An emotionally damaged historian tries to develop a chronoscope to look back in time, giving rise to an intriguing exploration of governmental suppression of the sciences.
--"Living Space"—Earth's too crowded, so they figure out how to transport to alternate universes (probability patterns) where each person has their own alternate Earth. This is quite a popular idea, until one planet-owner hears noises on his supposedly empty planet. Creepy and fun.
--"Profession"— People have forgotten how to learn from books and experience. Instead, all the knowledge for their indicated profession is zapped into their brain at age 18. This turns out to be a problem for the clever main character, who makes the mistake of studying for his dream profession before his 18th birthday.
--"The Gentle Vultures"—A race of non-aggressive aliens has been camping out on the moon, waiting for nuclear war on Earth, so they can go in, clean up, and restore a more peaceful society—for a fee, of course. They've been waiting and waiting, not understanding why nuclear war hasn't broken out, like it has on every other planet that reaches this level of development. Finally they decide they'd better start it themselves. One of my very favorite Asimov stories.
--"All the Troubles of the World"—Multivac—the mile-long computer—gets fed so much data constantly that it now predicts all crime that will happen, so the police can arrest people before they do the crimes. Now, for the first time, Multivac predicts an attempt on itself, and the Central Board of Directors races to neutralize the threat. Fascinating. Great ending.
--"The Ugly Little Boy"—Scientists pull a cave boy out of the past and study him. He, however, becomes more than just a test subject, which causes problems when they have to send him back to make space for other subjects. Touching, with interesting ethical debates.
--"Nightfall"—on a planet with so many suns it's never dark, scientists and sociologists predict a total eclipse, which will plunge the planet into a darkness which will drive the population mad.
--"Hostess"—A biologist—one of this collection's few strong female protagonists—discovers a sinister conspiracy involving Earth's relationship with the few known intelligent alien races. Rather ethically terrifying.
Discover this imaginative collection for yourself. Available here.