Newton's laws of mechanics fundamentally changed the world, yet they are relatively simple. The first and third laws seem very straightforward:
An object will follow a set velocity until a force acts on it, in an inertial frame
Every force has an equal and opposite force
I understand that the second law has a lot of calculus baked into it, but even then, the idea of how something accelerating depending on how heavy it is and how hard you push it is also pretty basic intuition. I also understand that air resistance might make figuring the first law hard, but there are thousands of years' worth of philosophers who never wrote down these relatively basic set of laws until Newton.
Were there any recent discoveries or precursors that led to Newton being the first to understand the laws of motion? Were these laws really impossible to understand without calculus, even in concept? Or did other philosophers know about them, but just didn't bother to write them down?
What events lead to Issac Newton being the first man to write down the laws of motion?
None, since he wasn't the first to write them down. He didn't take credit for them either. In his Principia, he notes after giving "his" laws of motion:
I have laid down such principles as have been received by mathematicians, and are confirmed by abundance of experiments.
That is, he says that they have been discovered before, and experimentally confirmed. He credits the first two to Galileo (or, at least, says that Galileo used the first two), and the third to Christopher Wren (who was a physicist as well as an architect), John Wallis, and Christiaan Huygens. While Newton says that Huygens was the last of the three to publish the third law (which he was, publishing it only in 1669), he was the first of the three to discover it, in the 1650s (Wren followed in the 1660s).
Today, Galileo is usually credited with the 1st law (inertia), but we can note that he considered uniform circular motion to be "constant" in the 1st law sense (which makes circular celestial motion "natural", not requiring any force such as gravity to maintain the circular path). Isaac Beeckman improved Galileo's work on inertia, but assumed uniform circular motion to be inertial. The correct version of the 1st law was by René Descartes and Pierre Gassendi.
While Newton credited Galileo with the 2nd law, Galileo's use of it was implicit. It was explicitly stated by Descartes and Huygens.
A modern phrasing of the Newton's 3 laws of motion is simple and short: "Momentum is conserved". With this in mind, we can see that Descartes was very close to the three laws with his "conservation of motion". However, Descartes' "motion" was mass times speed, not Huygens' and Newton's mass times velocity. Almost, but not quite, momentum.
Newton could have claimed more credit than he did. He appears to have been the first to use the third law with action-at-a-distance forces. Huygens had only considered contact forces. However, Newton wanted to avoid any debate on the correctness of the laws of motion. These weren't what he was interested in claiming credit for - his goal was credit for his law of gravitation.
For Huygens' early work and the conservation of momentum, see