Q:

How did humans learn how to kill Titans?

In Attack on Titan, how did humans learn the Titans' weak spot? How was it that they learned how to kill Titans?

Was it through experiments with live, captured Titans?

The only example I could think of where humans successfully captured live Titans and managed to get them within the walls was when they captured Sawney and Bean. But, during the time they had Sawney and Bean up until their untimely deaths, humans were already actively aware of the Titans' weak spots, which was the back of their necks (which could be cut/sliced/attacked).
I am rewatching Attack on Titan now and am currently on Episode 1. I may have overlooked something the first time I watched (hence I am rewatching it) but I am quite positive this might not have been answered.
So how exactly did humans learn/discover how to kill Titans? I still don't quite understand this. Manga answers are perfectly acceptable. ;)

A:

Your question is partly answered in the 17th chapter of the Manga SHINGEKI NO KYOJIN - BEFORE THE FALL.
This answer is a SPOILER for anyone who hasn't read the manga after chapter 16.

 In this chapter, Instructor Jorge recalls the last Survey Corps expedition before their dissolution, before the beginning of the current story. He recalls the tests done on the captured titan. These tests of theirs sealed with finding the weaknesses, if any, of the titans, by hacking into their bodies with weapons made of Iron Bamboo. They discovered that their lower part of the body (limbs), and the head wasn't their weakness. Then they were interrupted by the onset of other titans from the South and the West.
 Here, the chapter ends and as the next chapter hasn't been released yet, we don't know how they got to know about the weak point. 
 What we do know is that Angel Altonnen killed Mammon (a titan) using The Device (a prototype of 3DMG) and discovered the weak point.

I will update the answer as soon as more information is revealed....

A:

The titans can sometimes be killed with canons if they hit the weak spot. It would be reasonable to presume that while they could not kill enough with canons to due to the regeneration, they could see the occational fatality and pass the information on.  Some inventive mind would then create a more efficient way to target that spot. I believe after the battle for Trost they specify how many are killed by canon. But don't have a means right now to look it up.

Edit:  In manga chapters killing titans by shooting them with shells is explicitly mentioned by Armin in the battle of Trost and in one of the diagrams discussing the different between grape shot and regular shot.  In addition:

 We learn there are individuals who erased memories of all other humans.  It is very possible that nobles with more information about the origins of titans revealed this info to those with the erased memories while holding other information back.

