The Delilah theme has been used as an example in the screenshots below, but this applies to all themes.
To change all colors, go to your Blogger dashboard > Theme > Customize > Advanced. The first section is "Links". You can click on "Links" to expand the dropdown and go to other sections to change more colors.
Click on "ADD" from the section of the color you want to change. A little window will pop up where you can change the color and you can click on "+" (Add custom color) to add a new color. You can refer to the screenshot below.
Replace the color in the Hex field with your own. Make sure you add # before the color (correct: #000000, incorrect: 000000 or ##000000). Then click on "Save".
This is what you'll see after you click on the "+" symbol to add a new color.