Social skills are broadly made up of three things:
Observation (body language, how people look at you, each other)
Listening (what people are saying to you, the tone, speed, variability of pitch)
Speaking (how you say something, etc)
Body language can be hard, because no-one has really codified it in an accessible way. Yes there are books, but they tend to drift off into science noise, or babble.
The problem is that we are used to watching actors. The problem with actors is that they (like most people) can't control their body language entirely. So what you see as affection on TV, might not actually be Affection. (this is why its so startling to see real affection in the movies. the so called "chemistry")
solution: at a cafe, or a shopping mall, sit with a set of headphones on, and just watch people (no don't stare) look at the parent with their children, the young couple, the old grumpy farts. nearly divorced boomers. Try and guess what they are thinking through their actions.
Where are they looking when they speak?
are they hunched over?
where is thier body facing?
Are their arms crossed over, or wide open?
Listening, again at a busy place, a food place, cafe or somewhere where there are lots of people, listen in (again headphones are a great foil for this sort of thing). What are people saying to each other? what reaction does it have? when people are happy, what speed is their speech at? is it all one tone?
speaking: An excellent tool for when you are not sure is "mirroring". That is when someone makes a statement that you are not sure of, repeat the last few words back to them: "hi OP, how are you? can you come with me please?" you: "come with you?" (the tone of your voice is important here. you need to be light, fluffy. subtle rising intonation helps) They will normally give more information "Yeah, I have something to show you" etc etc etc.
A good rule of thumb is that the words people are speaking only contain something like 20% of the meaning of what they are saying. Think of the ways you can say "are you having fun?" The delivery can change the meaning from innocent, genuine question, menace, anger, sadness. Even though the words are exactly the same.
finally the last thing you need to work on is "empathy". No its not a wooly thing. Its the ability to predict how the person you are talking to is feeling. it doesn't involve crying. basically its all the information you've gathered, condensed into a model, and use that model to understand why they are acting the way they do. hint: people are never irrational to themselves.