Let me give you a couple pointers; I did my Master's thesis in quantum transport.

(A) Measurement and observation is NOT a simplified term for a well-understood complex process. It is an atomic term for a not-well-understood (therefore maybe simple or complex, we don't know) process which is really extremely simple if we take it at its surface meaning and don't poke inside it too far.

Let me take the simplest example, the Stern Gerlach experiment. This is really simple: put two long magnets next to each other with a "gap" between them, and preferably give them very different shapes; this creates an "inhomogeneous magnetic field" between them. It turns out a spinning electric charge, when it comes into such a field, should get deflected based on the direction that it's spinning.

Fire a beam of electrons at this apparatus, and you'll notice something interesting. Let me give some coordinates: if the gap between the two magnets is "horizontal" and the electrons go through it "forwards", they split into two beams, half going "up" and half going "down". We say that these have 'spin up" and "spin down" but that depends on a bunch of little arbitrary choices. If you put another apparatus "horizontal" in front of either beam, you'll notice that those ones going "up" all go "up" through the second set of magnets; likewise for the ones going "down".

So there's a lot to unpack here: first off, if they were normal spinny things, then this separation into two beams is really weird! Because what about an electron that's spinning "forward" or "left"? Why wouldn't nature recognize that mathematically it's spinning just as much clockwise as anticlockwise, vertically, and not deflect it up or down at all? So there should classically be lots of particles deflected between these two beams: it is very strange that this does not happen! In fact we can perform the experiment. We can use a second Stern-Gerlach magnet, this time oriented vertically, so that it deflects electrons into 2 beams going left or right, call these "spin left" and "spin right." Now we take the spin-left electrons and first put them through a vertical Stern-Gerlach magnet, make sure they keep going left, great. We just rotate the magnet and we find half of them go "up" and half of them go "down." Nature doesn't know the difference between "left" and some sort of 50/50 mixture of "up" and "down"; those are the same to Nature, at least where an electron's spin is concerned. And that finding is very robust: "up" is a 50/50 mix of "left" and "right" so if you use another Stern-Gerlach magnet on the electrons that went left and then up, you do NOT see them all go left again! They will all go up if it's horizontal, but they will not all go left if it's vertical. Instead half of them go left and half of them go right!

Now, we have a very folksy understanding of what we mean when we measure these things: we stick a very sensitive electron detector in some place, connect it to a counter, and we watch the counter erratically tick upwards. Of course it is so sensitive that it ticks upwards due to all sorts of other noise sources, even without a signal, but when we turn on the electron gun we start to see that in some places, pointed at the place where the beam hits the gap between the magnets, it starts rising much faster than the noise would provide, and in some places it doesn't rise any faster at all; it's all attributable to noise. That's how we know there are these two beams coming out; we move this detector around and see some peaks in the detection rate.

Now, a lot of our explanations of how the electron can do all of these weird interactions with the magnets, depend on saying that the electron does not just take one path at one time! Instead maybe it is "spread out" in space or it "takes all the paths available to it" or something -- these funky interpretations make the rest mathematics used to describe the electron unbelievably simple, you just have these "unitary transforms" and this "linear evolution" and all of that complicated quantum mechanics stuff is super-simple mathematically. But when we measure, we just see this counter jittering upward with highly unpredictable increments but with some very predictable average rate. Most of those clicks we'd like to think are actual electrons which have made up their mind to take this path or that path and have successfully made it to the counter. How this happens, is something of a mystery. If the electron takes all paths, why does it end up here or there? If it's spread out so that it's half on this detector and half on that detector, why do we see the counter increment and not, say, fuzz between the two numbers, having half-incremented and half-not? Why isn't our world more fuzzy, if it's made out of these fuzzy probabilities and amplitudes at its core? And yet why, when we use scanning tunneling microscopes, do these electron clouds of these atoms look like little balls, as if those electrons really aren't spread out over all that space but occupy one single place all of the time?

The mystery comes because there's a lot of really easy ways to explain this funky Stern-Gerlach stuff, but most of them view the world in a way that's alien to our own. The measurement problem is "we know how measurement works pragmatically, and it never showed us this alien world before, so how is this alien world 'collapsing' into the familiar world that we all know and love?"

(B) Entanglement has to do with strange correlations between remote systems which you can only notice when they are brought back together. My favorite example is a game where 3 people compete as a team in several trials where we secretly put them at cross-purposes to each other, call it "Betrayal." We split the team of 3 people into 3 separate rooms and we prohibit communication between team members. Each room has a screen that we display a goal on, and two buttons labeled 1 and 0. Sometimes the displayed goals for an individual and the actual goals for the team will be at odds; the individual never gets any reward in these cases: it's only, "if the team gracefully recovers from our meddling 100 times in a row, we will give them all a big cash prize."

Okay, so how do these work? Once the people are settled in their rooms, 1/4 of the time we will broadcast a "control round" where we tell them all "make the sum of your three button-presses even," and start a countdown timer. They win if they all push exactly one of their buttons once before the time is up, and the sum of their pushes is even. Really simple. Then 3/4 of the time we will choose one of them at random to be a "traitor" to the other two: we tell the traitor, "make the sum of your three button-presses even," but we tell the others "make the sum of your three button-presses odd," and the team wins only if they each push exactly one of their two buttons once, and the sum is odd."

It's easy to prove that you cannot win this game more than 75% of the time classically; each of the 4 situations is represented by some equation among the 6 correlated random variables, but when you add all 4 equations together you find out that they reduce to 0 = 1, an obvious contradiction, so they can't all be simultaneously satisfied no matter how you correlate the random variables. It is also easy to prove that if they all start out with a class of entangled states called a "GHZ state", such as

|+++> + |---> = |000> + |011> + |101> + |110>

then they can either measure their state to get an even sum, or any two of them can perform the unitary transform mapping |+> to |+> and also mapping |-> to i |->, yielding the state

|+++> - |---> = |001> + |010> + |100> + |111>,

and then any measurement must yield an odd sum. The two who are told to make the sum odd can do this with absolutely no help from the one who does nothing to make the sum even. In theory, the only limit to your accuracy is how long you can keep these GHZ states away from outside noise and disturbance. And of course we can account for that by only requesting that you pass, say, only 90% of the trials successfully -- with enough trials we can still prove that a classical team with their 75% upper bound on success in individual trials will almost always fail whereas a quantum team whose tech is good enough to get to a 95% success rate will almost always pass enough trials.

But, you can't use this spooky collaboration to transfer information faster than light. And that's precisely because you can't discover that the two measurements are correlated until you compare them! We instantly correlate but we aren't instantly aware of our correlation; I can't figure that out until you send me a message saying, "hey, is your set of numbers X?" and I say "yes it is! woah! spooky!"