Here's a quick guide to painting American Civil War Union & Confederate forces. My experience has mainly been with 1/72 scale and 1/32 scale plastics but there is no reason why at least some of this info should not be of use to those painting other scales or materials.
The paints I have used for years are water based acrylics from the Vallejo Game Color range. For the ACW the colours I use most often are:
#10 Bloody Red
#20 Imperial Blue
#21 Magic Blue
#23 Electric Blue
#40 Cobra Leather
#41 Dwarf Skin
#45 Charred Brown
#48 Sombre Gray
#49 Stonewall Gray
#50 Cold Gray
#56 Glorious Gold
#63 Desert Yellow
These 16 paint colours cover most common eventualities for both sides.
(Although the more I think about it the list of exceptions grows (!) with the likes of Berdan's Sharpshooters, artillery pieces, limbers & wagons, some Zouave units and other more exotic early war regiments. You may also need a yellow if doing cavalry.)
This link takes you to the product listing in my store if interested. Note that ACW 10 and 4 bottle sets are also available if 16 is too many:
By mixing the paints you can add almost infinite varities of colour, tone and shade. Bear in mind also that even for the much more uniform look of the Union side there would have been considerable variation in clothing colour. It would have been made in numerous different factories and also changed colour as it was exposed to the elements on campaign.
So there probably is no absolutely "correct" colour for anything - just what you are happy with having done your research on the likes of Google and in books such as the Osprey range etc...
Below are some examples of 1/32 scale ACW figures from A Call to Arms that I have painted using Vallejo paints.
Iron Brigade (Set #18)
Union Colored Troops (Set #11)
Same poses, different colour trousers.
Union Western Theatre "Bummers". (Painting conversions of Confederate set #16)
"Butternut" Confederates (Set #16)
"Gray" Confederates (Set #16)
If 1/72 scale is more your thing then these two links showcase my ACW Union and Confederate armies which like the big chaps above were all painted by me using Vallejo Game Color water based acrylics and Javis brushes.
Some makers such as Imex, Italeri and Revell helpfully offer painting guides on the back of their figure box sets. I've featured some of these below. They suggest their own preferred paint ranges but it is not that difficult to transpose these to your own paint range if different.
Italeri Union Infantry Set #6012
Italeri (ex-Esci) Union Infantry Set #6177
Italeri Union Cavalry Set #6013
Italeri Union Artillery Set #6038
Italeri Confederate Infantry Set #6014
Italeri (ex-Esci) Confederate Infantry Set #6178
Italeri Confederate Cavalry Set #6011
Imex Union Cavalry Set #503
Imex Union & Confederate Artillery Set #601
Imex Confederate Infantry Set #506
Imex Confederate Cavalry Set #504
Revell Union Infantry
Revell Confederate Engineers
I am happy to try and help if you have any queries - message me.