I love to read articles by artists about their palettes. There are so many approaches to color mixing. As long as there is harmony I don't believe it matters whether an artist uses many colors or just a few, but limiting the number makes harmonizing easier, especially for the beginner. An added benefit of using only a few colors is the ease of taking inventory and restocking.
The simplest palette is one of each primary and white. There are more mixing options with a warm and cool version of each primary plus white, such as ultramarine blue, Prussian blue, alizarin crimson (cool), cadmium red (warm), cadmium lemon (c), cadmium yellow medium or deep (w), and white. I usually use this latter choice (but often with ultramarine as my only blue) with one or more "convenience colors" thrown in. I like cadmium orange because I cannot mix an orange with this much staining power and I use it regularly to gray my blues. I also like to include an earth tone such as burnt sienna or raw umber to gray other colors, mix earth color variations, or add to ultramarine blue to make darks.
That said, I always use the palette recommended by the instructor in workshops so I can experiment. Whenever I find interesting articles about color mixing and palettes I save them for future reference. Here are a few that have been especially helpful or interesting.
Terry Miura recently posted this on mixing greens and I encourage you to read it. http://terrymiura.blogspot.com He lists the colors on his palette- a warm, cool and low chroma (grayed down) version of each primary plus white. He goes on to discuss how he uses these colors to mix different greens, a problematic color for the landscape painter. He uses the same palette for his other subject matters as well.
Kathleen Dunphey paints majestic west coast landscapes/seascapes but she also paints still life and animals. Her palette is quite limited (just 6 including white) and she lists her colors in this post- https://kathleendunphy.com/blog/32062/keeping-it-simple-using-a-limited-palette Take a look at her choices. She lists which brand she uses for specific colors (there can be a big difference between brands). I have worked with this palette a bit and have found it to be very versatile.
Thomas Jefferson Kitts has written a great article regarding the various plein air palettes of contemporary painters. http://www.thomaskitts.com/2013/04/the-contemporary-outdoor-painters.html You can see what many of your favorite artists are using in the field.
Try this challenge- paint one (or more) paintings using the Zorn palette:
white, black, yellow (usually yellow ochre) and red. That's it! Black is blue when white is added, yellow and black make a great green. Anders Zorn was a Swedish artist. He didn't always use such a limited palette but was well known for it. Look up his work if you aren't familiar.
Here is a scene I painted in the studio using titanium white, ivory black, cad. yellow light and cad. red medium.
I didn't have trouble mixing the colors I needed but I missed the transparent darks I can get with my usual palette. With practice I could probably get the hang of thinning them down. This palette lightens the load for outdoor painting and travel. Color harmony is a given.
And last, regarding all those tubes of weird colors that you never use and don't remember why you bought them in the first place, try adding only one at a time to your usual palette to see how it mixes. You might make new discoveries and you'll use up some of that paint!
I have barely scratched the surface of this huge subject but I hope I called your attention to some helpful links.
Keep mixing and experimenting!