Won't happen with gases. Not if you want Terran type life in the open. The blue of our sky is primarily Raleigh scattering. https://en.wikipedia.org/wiki/Rayleigh_scattering
This is going to be the bluish end of whatever light the atmosphere is transparent to.
Most gases aren't coloured. Exceptions:
* Chlorine -- yellow-green
* Bromine -- brown
* Iodine -- purple
* Nitrogen dioxide (NO2) -- redish brown.
But all of these are toxic.
Table with more colours: https://en.wikipedia.org/wiki/Color_of_chemicals
A better way to colour sky may be with dust. Mars's skies are pink due to superfine dust floating in the air.
Most of our earth dust is white or grey, so you get dull colourless skies.
Smoke is fine enough that it shows some Raleigh scattering -- hence bluish colour when lit from the front. Mountain haze (mostly organics and water) is bluish too. Lit from behind the colours are redish. The blue got scattered, and red is what's left. The more particles there are, the dimmer the light, which you perceive as being more saturated.
Gold, if fine enough (A few atoms thick) will be green in transmission, and gold in reflection. So around the sun in the sky it would be green. Away from the sun it would be gold. Raleigh scattering would add a blue component at right angles to the sun. The world would overall have a greenish look to it from the filtering. You need some interesting geology to create this.
In general any colour of dust should be usable this way:
If the particle size is large enough that it is opaque, then reflection is the only activity. The colour of the dust dominates, but is mixed with raleigh scattering.
If the particle size is small enough to be transparent, then you two colours -- the bulk colour as back scatter, and the sunlight minus the bulk colour as forward scatter, again with Raleigh scattering off the gases.
If the particle size is small compared to light, you get Raleigh scattering off of the particles too.
In addition to this, particle shape has an effect. Ice in Earth's atmosphere produces a bunch of effects from refraction, reflection and diffraction. Start here: https://en.wikipedia.org/wiki/Halo_(optical_phenomenon) With coloured crystals you could tint the resulting phenomena. Note that the angles mentioned in the article are specific to water ice. You may need to get more detail off the physics forums.
You're going to need to do a bunch more research, but you have to potential to put on quite a show in your story.