Motivation for doing classical mechanics This question is based on the philosophy of this, and answers should be flavoured in the answers of usual motivation tagged questions on MSE.

I know calculus, but to be honest, I don't know any physics. I plan to learn physics, beginning from the classical mechanics viewpoint - but what's demotivating is that in the beginning, everything is seeming to be an uselessly formal version of intuitively obvious things. It's quite obvious from common life intuition how springs will behave, how a car will go, or why you feel heavier or lighter going down the lift. 
(Also the problems in the textbook are stupidly boring - I don't think anybody cares about complex configuration of masses in a pulley, or, say, a ball hitting some other balls with some velocity - so I need something cool and counterintuive to analyze by myself.)
Of course, my viewpoint is heavily mistaken - there must exist counterexamples (which should make CM interesting to study) - but when I'm starting to study, I'm not finding any. 

Can you provide (everyday life) counterintuive phenomenons which can be explained properly with classical mechanics ?

 A: I believe that the gyroscope is the most tricky and misterious thing in the universe. Black holes? Entanglement? No biggie... But the gyroscope? that I can't yet understand.
To provide you with more motivation to do Classical Mechanics besides the counterintuitive phenomena such as the gyroscopic effect, Coriolis Force, Magnus Effect (actually fluid dynamics but it's classical mechanics too) let me say this:
When one makes a "uselessly formal version of intuitively obvious things" as you said, one eventually find deeper meaning on things. For example, the fact that (almost) every equation of motion of a classical system can be derived by minimizing (or maximizing) a really deep abstract thing called action is astonishingly beautiful and mysterious.
Even more, sometimes this insights are so deep that they provide evidence for new physics. Following the example, when one developes Quantum Mechanics one finds out that actually the action is just the phase of the probabilistic weight of each possible quantum path a particle can take and that the classical path is just the most probable one (this is just an enormus simplification).
A: So I second P.C. Spaniels argument about the gyroscope.  I'm pretty sure gyroscopes are actually aliens, and that's after watching the brilliant VSauce video on the topic.
The reason it seems like all physics does for you is formalize intuitive things is because you're just starting.  We intentionally start with intuitive things because it is more comfortable for people.  We could start with the surface effects of ferofluids, or flow separation over turbine blades, but why confuse people?
Your intuition will get you far, especially if it's good, but at some point you're going to hit limits.  At some point you're going to want to have the formalisms to help you convert the things you are looking at into math.  An example where this intuition limit started hitting for me is wave mechanics.  It's not so easy to explain why this flow of water goes backwards, or what patterns are going to appear next on a Chladni plate.
Eventually you can even get into exotic environments like orbits in space, where if you want to catch up with someone, the best way to do that is to use your thrust to push yourself towards the center of the Earth, rather than directly at the object.  Why that's the case is easy to explain with some basic orbital mechanics, but good luck understanding.
Or maybe you're interested in how cats always land on their feet.
So don't worry.  The physics will get counterintuitive soon.  Until then, we're trying to build up your trust in the physics.
A: I remember the first lecture on classical thermodynamics where our teacher asked if anyone knew what temperature actually is. At first you think this question is ridiculous, but then you realize you haven't the slightest idea. I found this to be a very pleasant moment in my education. Similarly, many other basic thermodynamics quantities are interesting to contemplate, with entropy definitely being one of them.
Also, with some linear algebra you can work out one of the most beautiful classical mechanics riddles I came across (presented to me by my classical mechanics teacher): take a frictionless 1D "plane" with a single wall. Place two masses $m_1$, $m_2$ on one side of the wall, and let $m_1$ travel at initial speed $v_1$ towards the stationary $m_2$ that is placed between $m_1$ and the wall. The masses will collide, sending $m_2$ towards the wall and rebounding back towards $m_1$. The super-gorgeous question is: how many times will the balls collide as a function of $m_1$, $m_2$, $v_1$? Collisions are completely elastic, no friction, all mechanical energy is conserved. It's not "counterintuitive", but it's a demonstration of great beauty in the work of precise mathematical tools in a physical context.
Finally, I think non-inertial reference frames present many counterintuitive scenarios, such as the Coriolis force, balloons in accelerating cars, pendulums, motion of icebergs and so on.
Additional rigid-body examples:


*

*Suppose you have two barrels with the same mass rolling down a hill. One of them has the mass concentrated close to its major axis and the other has its mass more spread out. Which would roll down faster? more generally, what is the optimal mass distribution to make the barrel roll fastest?

*You're driving a car uphill (without slipping, just driving normally). To which direction does friction force works upon your car?
And to wrap things up with one of most famous classical mechanics problems of all time: the brachistochrone: suppose you want to design a slope such that a skier that slides without friction and only due to gravity will reach the endpoint in the shortest time possible. What is the curve that achieves this goal? Rephrasing, what is the "fastest curve" you can make? Is it a straight line? A quarter-circle? How should one approach this problem?
