Listen to some of the old Carter Family recordings. Mother Mabelle played a lot of the melodies on the bass strings with a thum pick, while doing fills or stums with finger picks banjo style.
Arlie, thanks for responding to my question, and thanks for the link. It was really helpful.
By definition, you can't play chords and melody at the same time : to make it simple, chords are notes packed together ("vertical reading", because these notes are represented stacked one upon the other on a sheet music, same on a tablature) whereas a melody is a series of single notes ("horizontal reading"). However, on the guitar you can "cheat" in a way by playing the melody through chords : chord melody as mentioned on other comments, playing up to 3 sound patterns (sorry I don't know how you that in English, plans sonores in French) using alternately bass, melody and accompaniement etc etc. All of these techniques are to make illusion as there were more than one instrument, but you'll be playing just one guitar! ;)
It's mostly used in fingerstyle guitar, and not only classical, but also jazz, flamenco, bossa nova, country, ragtime, celtic, blues...... And you can also do similar techniques with a flatpick (and some even combine flatpick hold with index and thumb while playing with other fingers of right hand!). I guess you should begin by chord melody technique if you're beginner because it's more simple, natural on the right hand without any complex technique requiring indepedence of each right-hand fingers. The difficulty would rather be on left hand since you'll have to learn chord patterns (and sometimes harmonically unsual or complex chords in order to make the tune sound "melodic") and also rhythm (bossa nova rhythms for instance are very syncopated, not very easy to learn, but worth the work). Then, the trick here is, to alternate bass and chord, accentuate on some note etc, in order to make melody pop up through accompaniement.
Classical guitar is indeed the most complete school for fingerpicking, but it requires a lot of discipline. But most of these techniques can be learn through other styles and guitars. To my mind, one of the best examples to illustrate your topic, is Prelude n°1 by Brazilian composer Heitor Villa-Lobos. This fabulous piece combines melody and chords in a bewitching way. Maybe not of your level (yet), but maybe it'd give you the will to study techniques!
Good luck and happy playing :)
Here is my couple examples of playing melody and chords together :-)
I've been playing chords only (aka accompaniment) for many years. When I became more or less fluent with chords, I started adding melody notes and accents to the chords. I would say that playing melody and chords simultaneously means the same as playing "advanced" chords, unless you start playing melodies very fast - much faster than the chords changes.
VB, I enjoyed the songs. Your "Jingle Bells" variation sounded a little like Spanish guitar. A "Flamenco Jingle Bells" maybe? It certainly sounds better in a minor key.
Thanks Rosemary! it's a good idea to play it a little bit like Spanish guitar and in some different styles. Probably, I'll try it next Christmas :-)
BTW, good step forward after becoming more or less fluent with chord changes would be to start playing notes from the simple major-minor chords as arpeggios and adding missing 1st, 3rd and/or 5th notes from the chord into those arpeggios. It will help to develop your ear and link the sounds with the chord patterns and fingerboard.
After that, it might be useful to start adding alterations - 4th, 6th, 7th, 9th, 11th, 13th, playing diminished arpeggios etc.