The most interesting thing for me is the way that the Mishnah paints a picture of life at the end of the Second Commonwealth of Israel/Yehudah, before the Romans cast the Jews into exile. It depicts the daily lives and customs of the people - not in the form of a novel or biographies (although some individuals are mentioned), but in a snapshot of what they did, and the rules which governed their lives can tell us a lot about them. Of course, these aren't just 'quaint' accounts, as some non-Jewish commentators have said - these were (and are) the laws by which we are required to live. In that respect, they are more than just our history - they are our heritage as Jews, for our generation, and for all those generations yet to come.
It's amazing to read how Choni HaMa'agal stood in a circle and argued with Heaven until it rained - not too much rain, and not too little, but the right amount to do good - this shows that a righteous man had real strength in those times when people seemed closer to Hashem. Even the head of the Beis Din (the Rabbinic Court) said that he ought to punish Choni for arguing with Heaven, but couldn't if Heaven chose to publically answer Choni's prayers!
It's mind-boggling to learn in Masechta Eruvin ("Boundaries") that a Camel can be used as the side-post of a gateway which marks the boundary of an Eruv (a shared space in which carrying is permitted on Shabbos) - even if the camel might die and thus no longer stand upright!
It's even staggering to ask whether a Golem (a roughly man-shaped and sized creature made from clay, and animated by Kaballistic power) can be counted as part on a Minyan (the 10 men required before certain communal prayers can be recited). [No, it can't...] and it all hinges on whether it has a soul...
It's very confusing trying to understand which objects can become ritually contaminated, and which can't - Masechta Kelim ("Vessels") describes almost all of the day-to-day tools used in ancient times - it tells you a lot about how people lived, because their daily lives were defined by what they used, and how they used them. If you don't know what an oven looked like, or how it was made, the laws wopn't make much sense. But read a good commentary (Mr. Posen of London has created a wonderful illustrated edition) and suddnly it all makes perfect sense!
It's techincally difficult to measure the Eruv boundary when there is a mountain in the way, but the method is to "span" gaps using sighting tools and measuring ropes. The gradient is accounted for by holding up a pole and then measuring from the top of the pole.
But one of the strangest discussions I can recall from memory is the one about building a Succah on a camel's back! (in Masechta Succah) A friend in Jerusalem once told me about a newly-obversant student at a Yeshivah who laughed at such an absurd idea, and drifted away from Judaism... until he went to India, and saw a hunting lodge on the back of an Elephant! Realising that even the seemingly absurd in the Mishnah is based on solid truth, his faith was restored, and he came back to Judaism!
Zeraiim - Moed - Nashim - Nezekin - Kodoshim - Tahorot
What is Mishnah? - Why is it so Interesting? - A Few of My Favourite Things - Mishnah Glossary