Voice-leading and counterpoint are the music theory terms for how make multiple melodic lines maintain independence and sound like distinct melodies. They are big and complex topics that are well-worth studying. But you can also use less-formal instrumentation, orchestration, arrangement, and production choices to help achieve certain effects.
For example, big, sustained, extended chords behind the melody can be a great role for things like hammond organ, string sections, choir, synth pads, etc.
Alternately, with "chimier" or more dynamic instruments such as acoustic guitar, piano, pitched percussion, harp, etc, it is pretty common to voice full-range accompaniment as repeating arpeggiated patterns. This helps a fluid, lyrical melody line to sit "on top" of the rigid triplet pattern or whatever.
The place where you tend to run into trouble is when a guitar or piano is plunking out quarter-note chords over an eighth-note melody... the melody tends to kind of melt into the chord and make it sound like you're just changing chord voicings (which can actually be a really cool effect, if it's the one you're looking for).
The terms for these things are:
voice-leading: the subset of music theory dedicated to the study of how chords and melodies interact. Looking at all the notes being played by all the instruments, and how they move through a whole harmonic and melodic sequence.
counterpoint: sort of the "OG" original voice-leading, specifically going back to choral music, and specifically dedicated to maintaining voice-independence. "Counterpoint" tends to imply that one is following the best and most formal rules of voice-leading, while "voice leading" can imply simply the study of what's going on in a rock or pop song, for example.
Instrumentation: what instruments are in the thing, what they are doing, and what their roles are.
Arrangement: related to all the above, the high-level decisions about whether this piece will be a simple voice-and-piano song, or a New-Orleans style ragtime jazz version with brass and marching drums, or a lavish pop production with strings and rock instruments and a gospel choir, etc. It's both about instruments and also stylistic choices.
Orchestration - kind of the more-formal study of instrumentation and arrangement, especially oriented towards a traditional western orchestra with SATB sections for strings, brass, and wind, plus percussion. Incorporates counterpoint and voice-leading, and also factors like tambre, loudness, and useful ranges of various instruments.
Hope that helps.