Especially if your interest is just a hobby try starting the the "open" chords: E, A, D, G, C - all major chords. From there you next learn the minor and dominant seventh version of those chords. So, you end up learning E, Em, E7, A, Am, A7...
While you learn those chord types you also want to start learning some harmony theory. I won't try to summarize harmony, but when you learn harmony with practicing chords you will find out why groups of chords like D, G, A or G, C, D work together. As you learn how to play more chords, you expand your harmony knowledge to learn about the minor and dominant seventh chords in groups like Dm, G7, C.
...Learning each song individually and memorizing chords for a particular song seems inadequate.
Your intuition is correct. When harmony knowledge is applied you learn abstracted patterns rather than concrete chord names. The music theory of harmony will give you the abstraction tools. Roman numerals is the usual way. So concrete examples like Dm, G7, C or Em, A7, D are abstracted to
ii V7 I.
The set of "open" chords I listed number only 15 (and they follow certain patterns, so it is 15 completely different things) but with that relatively small number of chords you can play a lot of classic rock.
At first you learn to just strum the chords, but lots of rhythm techniques are used to dress them up. Fingerpicking & flatpicking. Playing fragments of chords. Varying them with little slides, hammers, and pull offs. Basically, getting your fingers to dance around those basic open chords.
Barre chords comes after learning open chords.
Methods for lots of instruments usually teach chords and scales together. In the case of classic rock guitar the scale you want to learn about first is the pentatonic scale and how rock music relates that scale to the blues. Especially important in rock is learning to bend notes when playing from scales.
Diatonic (major & minor) scales apply to rock too, but in a more theoretical way. Chord theory makes lots of references to diatonic scales (basically, chords belong to major and minor keys or scales.) You will want to understand those scales, but you won't necessarily play them as much as pentatonic scales.
So, open chords, barre chords, pentatonic scales. Harmony theory applied as you learn. Fingering and pick techniques applied to give it all rhythm.
Obviously, this is only an overview and not an actual method. I listed out some important things that everyone learns. If you are going to study on your own, look over various methods for these elements and pick some that feel right for you. One method might organize the theory well and provide drills and patterns. Another might focus more on how things are applied in real songs. Etc. Etc. I would choose a few different methods to compliment one another. There isn't one perfect method.
Another thing you might do is list out some songs that represent what you want to be able to play. 10 or 20 songs. You can use them as actual performance goals or to guide the kind of methods to use. An interesting possibility in the Internet age is to take your list and look up some good tutorials for those songs at Youtube. Not to learn all of them right now, but to get an introduction into what techniques and theory knowledge goes into those songs. It might help to see a big picture review of a set of songs so you know what is important to focus on.