Back in the day, in the early 1980s, the best way to learn about electronics was through the Babani books. One of the first ones I bought was Babani book 58: "50 Circuits using 7400 ICs". I bought a 7400, a few LEDs, some veroboard, and an IC socket, and built many of the circuits in the book.
That same 7400 Quad two-input NAND is now on the Retro Shield and I have used it to play with digital logic along the lines of the circuits in the old Babani book, which was great fun. Some example logic circuits are provided here.
The 5V 74xx series of TTL (transistor-transistor) logic was one of the first general purpose sets of digital logic gates, and covered loads of different gate types and combinations; though the basic 7400 is by far the most versatile, supporting many digital circuit designs.
As a good way to learn about digital logic, get a hold of the Babani book, and a 7400 like that on the Retro Shield, and have a play!