Get ready for a butt-numb-a-thon.
This is what you should read to get caught up on the world of Dragonlance in the "Age of Mortals." I'm only going to list the 'core' books for now.
Dragonlance Chronicles consists of:
- Dragons of Autumn Twilight
- Dragons of Winter Night
- Dragons of Spring Dawning
Dragonlance Legends is made of:
- Time of the Twins
- War of the Twins
- Test of the Twins
At this point, if you want to learn more about Raistlin & Caramon's early days, read:
- Soulforge
- Brothers in Arms
Another optional, but recommended read, is
The Second Generation.
Now, the original 4th Age of the first stories ends, and the 5th Age begins in
Dragons of Summer Flame.
Skip ahead a bit.
Now, we come to events which turn the 5th Age into the Age of Mortals and brings us to the present day:
War of Souls
- Dragons of a Fallen Sun
- Dragons of a Lost Star
- Dragons of a Vanished Moon
All of these books are by the original authors, Margaret Weis and Tracy Hickman. These are the 'essential' books for Dragonlance (with noted exceptions). The three trilogies are the must-reads to really get into the setting.
This is why people compare DL to Middle Earth in scope.

There are tons of other novels by various authors, but these are the important ones.