Start here if you're new to reading non-fiction as well as philosophy.

What Looking at Philosophy is to the history of personalities and ideas approach, this is to the problem analysis and solving approach. If you can only have two, take this as your second choice.

Not as comprehensive as you might guess based on its bulk , Joad provides clear insight into a number of stances that may otherwise perplex you as you move further into the subject.

Presented as a series of articles on the basic fields of philosophy, this one will tell you most of the concerns/problems that contemporary philosophers grapple with in their research.

Similar to Grayling's Philosophy 1, but shorter pieces on the more "fringe subdisciplines" such as philosophy of psychology, religion, politics, mathematics, etc. Also used at the University of London

You also don't know philosophy if you don't know Aristotle. Everything I said above about the complete Plato applies here. Don't forget to pick up both volumes.

Philosophy has a sort-of cleavage between analytic and continental traditions. You need to know both. This is the best introduction to continentalism by far.