Figuring out which colors work together can be a giant PITA, especially if you're just going with your gut. Thankfully, humans have been thinking about color since forever and there are some simple rules that will take the frustration out of your next project.
In fact we've come so far that there is even a website that lets you choose a color (or enter your own) and it will tell you all the colors that go with it in all kinds of ways. It looks like this:
So let's say you really like pink; pink is awesome so nice taste there. Specifically you like this pink:
You can choose that pink and you'll get a dynamic display of a complimentary color scheme. Like this:
Now you have two different pinks and three greens to use. These will all play well together and the green will really pop because it's literally the opposite of pink (more on that in a bit).
You can also go with a monochromatic scheme:
Or any of these:
In any of the schemes you'll get a color palette of other colors as well as the defaults. Like this:
Note that they show the hexadecimal code for each color which is how computer's think about colors. You can copy those numbers into Photoshop and most graphics programs. Or you can use EasyRGB to find a paint color, they have a bunch of commercial brands listed and a general output that includes Pantone.
Why theses rules work
When I was studying photography I was amazed to find out the ways you can use complementary colors. A pink dress jumps out from a background of green grass and an orange ball from blue sky. Here are some examples of complementary colors:
Each of those colors are on opposite sides of a color wheel. That means when you mix them together you end up with no color: either grey or black or white. There are lots of rules you can use as you can see from the different color-scheme examples I showed above. Of course the whole point of this post and this tool is that you don't have to think about any of this, just pick the colors you like. But here's a link to color theory on Wikipedia if you do want to think about it some more.
As a bonus here's a color wheel from 1776 because humans are awesome: