I'm looking for an introduction to Karl Poppper's philosophical ideas that have influenced the world the most. I have no prior knowledge of his work.
The best second hand accounts of Popper are in chapters 3 and 7 of David Deutsch's book "The Fabric of Reality" or Chapters 1, 2, 9, 10, 15,16 of "The Beginning of Infinity" by Deutsch.
The vast bulk of second hand accounts of Popper are so bad as that it is difficult to believe they are supposed to be about the material they are supposedly commenting on. This includes almost all commentary by professional philosophers such as Lakatos and Feyerabend.
Popper's best contribution was to epistemology. Part I, Chapter I of Popper's book "Realism and the aim of science" explains what is different about Popper's position, especially the first two sections. "On the sources of knowledge and of ignorance" the introduction to his book "conjectures and refutations" covers the same ground from a more historical perspective; chapters 2 and 3 of that book are also good on epistemology. Chapter 1 of Popper's book "Objective Knowledge" is a good account of his refutation and replacement of inductivism. The first two chapters of "The myth of the framework" are also very good.
Popper's political philosophy is better than most political philosophy, but considering the competition that's not great praise. "The open society and its enemies" is long and has some problems but is okay. Chapters 4 and 18 of "conjectures and refutations" are very good on political philosophy. Many of the other chapters in that book are okay.
Popper is a true giant, and second-hand or condensed accounts will deprive you of the life-changing experience of reading him. As Schopenhauer wrote,
For the thoughts of those extraordinary minds cannot stand filtration through an ordinary head. Born behind the broad, high, finely arched brows from under which beaming eyes shine forth, [the works of genuine philosophers] lose all power and life, and no longer appear like themselves, when moved into the narrow lodging and low roofing of the confined, contracted and thick-walled skulls from which peer out dull glances directed to personal ends.
— Arthur Schopenhauer, The World As Will and Representation Preface to Second Edition
You have at your disposal a vast wealth of Popper's writings to choose from, according to your tastes and what problems bring you to philosophy. A lovely first introduction to the man and his spirit may be found in All Life is Problem Solving (2001).
Having encountered Objective Knowledge: An Evolutionary Approach (1979) somewhat late in my path through Popper's work, I found it remarkably accessible. I conjecture that it would make a good first encounter with 'full-strength' Popper, but I also can't be sure its apparent accessibility can't be attributed to all my earlier investments in his other works.
That said, if there is one second-hand account I can recommend without reservation, it is that of Bryan Magee in Confessions of a Philosopher (1999). Magee knew Popper intimately, and judged him the greatest philosopher of the 20th Century. But he also offers interesting criticisms both of his philosophy and his character, the wisdom of which I'll say I have only appreciated begrudgingly and in restrospect. (If you do seek out this book, try to find one of the early editions, e.g., in hardcover at a library. A libel suit forced the excision of some scandalous but truly interesting material related to a different 20th Century philosopher.)