If you are using standard tubes of acrylic colour (there are several brands, all of which are fine) or acrylic craft paints (in bottles) the process is fairly easy.
As burgundy is a red-purple colour you can start with a real purple and add red to it. I have found three parts purple to one part red is about right.
If you are starting with primary colours only then a mix of six parts blue to eight or nine parts red will get you a good burgundy colour. Remember to mix the paint well and use your eye to judge the colour, adjust with either red or blue as you see fit.
Really, it depends on which shade of purple it is. Maroon should be close enough to purple, since it's formed in the same way purple is-with certain shades of red and blue. I would say, maybe, a shade of blue, a shade that's not too heavy, but not anything lighter than sky blue or periwinkle.
Do you want a warm shade of maroon, or do you want it to be cool-something to consider.
Cool: Start with a primary red color, of course. Add a small amount of primary blue (you can always add more later, but too much will make purple), and then add black until you achieve the tone or value of maroon that you desire.
Warm: Same as above, except instead of the blue you would use a small amount of primary yellow to achieve a warm maroon that will have a hint of a brown color as its base (you can not add brown itself to it, as it is not a primary colour and will create mud).
I think white would look lovely
Pink in the same colour family would look nice, as would cream or a light grey.
Yellow and orange are the colour when the colour mixes to make the colour maroon.
red and a hint of brown/black.
Purple and red, I believe.
Dark Purple
red and blue
red and a hint of brown/black.
Red and Maroon.
maroon
Mostly red about 25% blue, a little bit of black should make a nice maroon.
Maroon
maroon
The school colors for the University of Phoenix are maroon and gold.
Mostly red about 25% blue, a little bit of black should make a nice maroon.
Maroon
The official school colors for Virginia Tech are Chicago maroon and burnt orange.