the 60s are kind of too far back to exactly remember (yes there's an old joke about that), but a few things do stand out. started out playing chords to accompany folk songs strumming with a flat pick. in answer to your question, the hard part is to define a specific and recognizable set of chords that only fit one song. many folk songs use simple and transferable chord progressions which are used on many songs, e.g. cfcg cfgc kind of thing. same thing with 12 bar blues progressions and doo-wop c-am-f-g thing. so something like 'abeline' sticks out because of it's more unusual eg#ae f#b7e verse sequence. another one is a lead belly ragtime piece, 'keep your hands off her' which i played (and still play) on both 6 and 12 string, progressing from strumming to flat-picking to finger picking as i got better. anybody else do 'walk right in'? i stopped playing with picks decades ago, and have played almost entirely finger style since.
on the rock side, one of the guys learned 'house of the rising sun' in all its arpeggiated glory (didn't even know what an arpeggio was back then), and that got even better when i started finger picking. now most of the folk and rock stuff was done with friends, so again i don't really have an exact answer to 'first song', mostly just milestones. for example, when i was teaching myself fingerpicking, i used the stones' 'factory girl' to learn claw picking and the beatles 'in my life' to arrange and feature what i knew at the time. both of these were done as instrumentals only. i still play a lot of old rock instrumentals like 'riders i the sky', 'apache', and 'pipeline'. based on an exchange with jersey tuning in this forum, i have begun 'walk don't run', but still don't have it all down. i recently got a compilation cd of old rock instrumental classics, and so i've been working on that one. or as jersey suggested, "venturing". if anyone wants to hear about my odyssey with 'sleepwalk', pm me. probably my most complicated arrangement is for 'maria elena'.
one last milestone. while searching for a 12 string back in the mid 70s, i was trying one out at a local music shop ('e.u. wurlitzer' when it was on bedford street (or was it west street?). i played 'stairway to heaven' as a test song. couple of guys in the shop clapped when i was done and asked if i wanted to join their band (i didn't; no regrets).
sorry for the longish and not so complete answer. much of the learning time is far in the past and i constantly revise the way i do things, so nothing is ever truly finished.