You mention classifying people into groups such as TAG, LAG, and calling station and that's a reasonable start. It will mostly help with pre-flop decisions and with narrowing down the possible range of hands that an opponent might be playing. But the difference between winning big, winning a little, and losing often comes down to close decisions in big pots on either the turn or river and that's where more detailed reads can make all the difference.
To go beyond simple classification of players, the first step is to take an active interest in every street of every hand, even if you're not involved. That takes a certain level of concentration that can be difficult when you're playing long sessions or not having a good time, but it can also be satisfying and almost fun: you're like a psychological detective, gathering clues and trying to piece together how people are thinking.
One of the first things I try to figure out about somebody is what "level" of thinking they are on. Are they thinking only about their own 2 cards and the strength of their own hand? (lots of new, fishy players fit this) are they actively thinking about what possible hands they are up against? (and are they good at it?) are they thinking about what other people are thinking of their hand? (now they're disguising the strength of their hand or setting up bluffs) or are they operating at even higher levels? ("I know they think I think they think I have this, so....), etc.
Oftentimes, especially with opponents you've only played a few hands with, there may be a big hand that they've played that sticks out--something that was played across multiple streets and ended in a showdown. These are the hands where you can really gather in the type of thinking that someone is using; in fact, any hand that goes to a showdown is going to be really valuable to you in developing reads, so grab a hold of those when they happen and mentally go back over exactly what happened in the hand so you can remember it.
Paying attention to bet sizes are incredibly important also. Someone with poor bet sizing is often a not very good player, but more importantly there's many players who give away information with the way they vary their bets, but it can be different for everybody. For instance, you may find that one person bets bigger than normal when they have a really strong hand, or that another bets really small when they have a big hand, trying to keep people in the pot. Some people might bluff smaller (less risk) and some might bluff bigger (more fold equity).
The last point I'd make is that I basically treat all new players as if they are playing straightforwardly until I see otherwise. Once someone shows they are "capable" of making some move, you can file that information in your mental profile of them. For example, once you see somebody make a bluff, you now know they are "capable" of it and have to take that into consideration when up against them (there are some players who never or very rarely bluff). Or maybe you see them make a bluff where they bet on multiple streets. Maybe you see them bet out with a draw into multiple players--some people won't ever do that but you now that this person can. Perhaps you see them make a call with a poor hand on the river--now you know that they're "capable" of calling with weak hands so you can value bet them stronger and maybe be less likely to bluff against them. Another example is that you find out somebody raised with a weak hand in late position before the flop--now you know they're "capable" of that and have to factor it into their range of hands. To tie in with what I said before, when you see any type of move (and especially when there ends up being a showdown), ask yourself now if that person did it because they were playing their own hand or because they were thinking about the other person's hand or basically: what "level" of thinking were they on for that hand.
As you say, "the rest can be learned over time", so taking the time to analyze your opponents and every hand that goes down at the table can keep you engaged with the game and make tremendous differences when you're faced with those big decisions that could go either way. There's actually not a lot of hands like that that come during a session, so any tiny bit of help you can get to make the right decision could have a huge impact on your bottom line over time.