A few other points of reference. I find pure white undyed yarn or fiber really daunting. I think, "I can make this any color in existence. Now choose one." I get overwhelmed and just dye it blue/green. I also don't love all colors. In fact I'm quite limited, as you probably have already noticed, in which colors I work with. I tend to stick with blue/green, burgundy/berry...maybe some brown. I stay very far from anything yellow. Orange is out too. I once explained to a nice woman in a spinning class who was pressing some yellow merino into my hands despite my protests, "I grew up in a yellow room."
"Oh," she said, placing the fiber back on the table, "I see."
So keeping my color issues in mind, here is my process for overdyeing.
Think of a color wheel.
Look at your yarn. See where it falls on the color wheel. Stick to colors that are neighbors (analogous) on the color wheel. Avoid the color on the opposite side of the wheel (complementary).* If you choose colors across the color wheel that is when you make mud. And remember you are not mixing paint. Nothing mixes when you overdye. You have your starting color and you are layering transparent color on top of it.
If you're worried you can build your color slowly, just keeping adding color until you like it. I would also suggest starting with yarn in a color you really dislike, something you will never use, because if you "mess it up" you haven't lost anything.
I use Mother Mackenzie's Miracle Dyes. They are for dyeing protein fibers like wool, mohair and silk. They need to have acid added to set the color. I, like most casual dyers, use vinegar. I'm not scientific. I just put a few glugs in the pot. If you want variegated, uneven color you can add the acid before or alongside the dye. It causes the dye to "strike" or stick to the fiber so the fiber will grab it unevenly. I like this look. If you want really even color let the yarn sit in the dye for a while before adding your acid. You can also loop yarn on a dowel, support it on the edge of the pot and cycle the yarn slowly and evenly through the dyepot for the most even color. I don't have the patience for this. I throw everything in my dye pot or crockpot (bought at a thrift store and used only for dyeing) and go.
You need to heat the yarn to set the color. I typically leave it on a low heat until the dyebath exhausts, when the dye has all attached to the fiber and no dye is visible in the water. If I'm able I turn off the heat and let it cool in the pot. This gives the dye additional time to set and also spares your hands from handling super hot, wet yarn. Rinsing the yarn when it's cooled also makes it harder to accidentally felt the yarn.
Dyeing in the pot will produce a lot variegation. The yarn at the bottom of the pot, close to the heat, will get the darkest while the yarn floating up at the surface will generally be the lightest. You can move things in the pot but be gentle so as not to felt it into one big ball.
When I dye in my crockpot I will leave it on overnight on low if I put the yarn in close to bedtime. Otherwise I shut it off after several hours and let it cool overnight.
So that's all there is to it. It's easy. You can always dye something again if you don't like it the first time. In fact, dyeing multiple times gives deep, interesting, complex color. There is nothing to be afraid of. You can do it!
Carol Sulcoski, who is a professional dyer, wrote a nice piece about overdyeing on her blog recently. You can read it here.
*Complementary colored yarns are great for colorwork projects. They really pop against each other.