It just comes with experience. There is no secret or rushing it. And it depends a lot on how much you play/practice. The people you see doing what you describe are likely professional musicians or have many years of experience to do that.
If you just play once in a while for recreation, it's going to be a long time coming to get there if at all. I used to wonder how it was done as well; not that I'm good at it either, I'm just better than I used to be. There are tuning tricks though that you should learn. Many pros can tune the strings to each other quite easily so you only need to know that one or more strings is already in tune. And a pro could hear another instrument or the radio or a CD and tune one note/string to that and go from there. I've never seen a pro who didn't have a tuner but if you know one or more strings are in tune, it's not hard to get the others close to dead-on.
As far as pickups/amps go, everyone's hearing is different. Don't feel that you have to or should like something because a lot of other people do. Keep in mind that your tastes will also change as you get older as will your discernment for different sounds. I see kids in the
music store who listen to some noisy horrible amp (to my ears) and they think it's great. And I liked amps when I was younger and look back and go "what was I thinking?". I think that in a given age group the opinions won't vary that much but I've never taken an official poll.
Let's say someone worships
Zakk Wylde or a metal band and loves their sound and thinks that EMG active pickups are the greatest things ever. For the sound they are going for they are right. But a professional musician who usually has to play a variety of things or may not be into that style of music will out of necessicy have much different tastes in pickups and amps. This is why pros usually have different guitars and amps as well.
Remember, if it sounds good to you, it is good until you decide differently. I can guarantee that your tastes will change over time though.