What are some good ways to form opinions on controversial subjects in the current and upcoming era?

Yes, it takes a crazy amount of time, so I choose my battles. In case of homeopathy and evolution I actually got the background knowledge regardless of controversy, as I was already interested in natural science. (Also, my wife is a biochemist, so I often defer to her expertise.)

For political/economical things, my rule of thumb is to listen to people who mostly do the "boring" work. Such as, collecting information on corruption, which 99.99% of time everyone ignores, and 0.01% explodes into a huge political scandal. When the scandal happens, I trust the opinion of the guy who worked on all the non-scandalous stuff over anyone who only pays attention to the "cool" stuff.

Basically, you need to know what is "the business as usual", and compare to that. For example, most people enter politics in order to make some kind of personal profit. But most of them stay within certain lines, and some of them go way beyond them. So instead of "did X embezzle money at Y", a better question is "did X embezzle the usual amount of money at Y, or did something extraordinary happen here"? If you ask the right question, you will see the greatest criminals. If you ask the wrong question, your attention will go where the media want it to go. Yeah, in a perfect world all crime would be prosecuted, but we don't live in that world. (For example the problem with Trump is not that he is bad, but that he is bad beyond usual. Some people believe that introducing random change to the system is good. I believe most of the time it only makes the system worse.)

With math... well, there are topics that other people consider controversial, such as Cantor's diagonal argument or the Axiom of Choice. My approach is that instead of "joining a side", I want to get a better insight into how these things work, on the mechanical level. (LOL, you probably know about these things much more than I do, but telling me the answer wouldn't help me much unless I can derive it myself. And that's kinda my point: I wouldn't spend my effort on trying to agree or disagree with you, but rather on trying to get to your level, at least in some very narrow area.)

The method of "put the numbers (that you can enumerate) in a table and invert the diagonal" provides a number that is outside the original set, which can be used in multiple ways. Like, if you put the rational numbers in the table, the inverted diagonal will be an irrational number. If you put the computable numbers in the table, the inverted diagonal will be an uncomputable number. So even if someone claims to be coming from a finitist or whatever other perspective, my intuition would be like "you guys basically redefined 'number' to mean X, and I am not going to argue definitions here, but anyway, when you put all Xs in the table, the inverted diagonal is going to be a non-X".

The real fun, from my perspective, is Skolem's paradox. I think I approximately understand the point. The first-order logic cannot express certain concepts (even quite simple ones, such as standard natural numbers). So you should not think of the axioms as describing "the Platonic world", but rather as applying to different models. (Feel free to think about the Platonic world as containing models of sets, rather than the sets directly.) Then instead of "wow, this is a mysterious paradox", you should approach it as a hacker and say "wow, there seem to be malicious models that technically follow the letter of the axioms, while completely ignoring the original intentions of their authors... sounds like a lot of fun". And then, whenever you hear a statement about something, you should be like "I wonder how this statement works in one of those malicious models". -- And, specifically, I wonder how exactly Cantor's diagonal argument works in one of those malicious models, but I am not there yet. I am reading a textbook that I have already read previously, except this time, I am like a hacker looking for a possible exploit. Like, when it says "natural numbers", I go like "I wonder what happens if those are nonstandard natural numbers".

(And I plan to approach the axiom of choice the same way. Like, the axiom is neither "true" nor "false", there are simply models that are compatible with it, and models that are incompatible, and I want to get an intuitive understanding of both.)

This will take me a lot of time, and perhaps I will never finish it in my lifetime, but that's okay because I enjoy math. :)

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data Bijection : Type -> Type -> Type where
  ||| A bijection between two types.
  ||| @ a the first type
  ||| @ b the second type
  ||| @ fwd mapping from first type to the second
  ||| @ bck mapping from second type to the first
  ||| @ fb_producer proof that fwd . bck = id
  ||| @ bf_producer proof that bck . fwd = id
  DoMap : {0 a,b : Type} -> (fwd: a -> b) -> (bck: b -> a)
          -> (fb_producer: (av : a) -> (bck $ fwd av) = av)
          -> (bf_producer: (bv : b) -> (fwd $ bck bv) = bv)
          -> Bijection a b

map_fneq : {0 a,b : Type} -> {f : a->b} -> {g : a->b} -> f = g -> (v:a) -> f v = g v
map_fneq (Refl {x = f}) v = Refl

bool_inv : {v : Bool} -> (v = not v) -> Void
bool_inv {v = True}  prf = uninhabited prf
bool_inv {v = False} prf = uninhabited prf

diagonal : (Bijection Nat (Nat -> Bool)) -> Void
diagonal (DoMap f g _ bf_prf) = bool_inv conflict where
  H : Nat -> Bool
  H i = not $ f i i
  
  h_hi_by_idx : f (g H) (g H) = H (g H)
  h_hi_by_idx = map_fneq (bf_prf H) (g H)
  
  h_hi_by_def : H (g H) = (not $ f (g H) (g H))
  h_hi_by_def = Refl
  
  conflict : f (g H) (g H) = (not $ f (g H) (g H))
  conflict = trans h_hi_by_idx h_hi_by_def

I misread a small bit, but I still stand by my answer. It is however still unclear to me if you value truth or not. You mention moral frameworks and opinions, but also sound like you want to get rid of biases? I think these conflict.

I guess I should give examples to show how I think:

• Suppose that climate change is real, but that the proposed causes and solutions are wrong. Or that for some problem X, people call for solution Y, but you expect that Y will actually only make X worse (or be a pretend-solution which gives people a false sense of security and which is only adopted because it signals virtue)
• Suppose that X is slightly bad, but not really worth bothering about, however, team A thinks that X is terrible and team B thinks that A is the best thing ever.
• Suppose that something is entirely up to definition, such that truth doesn't matter (for instance, if X is a mental illness or not). Also, suppose that whatever definition you choose will be perceived as either hatred or support.
• I don't think it's good to get any opinions from the general population. If actual intelligent people are discussing an issue, they will likely have more nuanced takes than both the general population and the media.
• Lets say that X personality trait is positively correlated with both intelligence and sexual deviancy. One side argues that this makes them good, another side argues that this makes them bad. Not only is this subjective, people would be confusing the utilitarian "good/bad" with the moral "good/bad" (easy example: Breaking a leg is bad, but having a broken leg does not make you a bad person). 

I think being rational/unbiased results in breaking away from socities opinions about almost everything. I also think that being biased is to be human. The least biased of all is reality itself, and a lot of people seem really keen on fixing/correcting reality to be more moral. In my worldview, a lot of things stop making sense, so I don't bother with them, and I wonder why other people are so bothered by so many things.

I might be unable to respond for a little while myself, sorry about that