Avoid using overly bright colours, a lot of people (particularly the older generation I've noticed) find them almost painful to look at on a bright screen.
I also usually try to avoid using pure black/white (#000/#FFF), since they sometimes come off too strong. I'm not saying they don't have their place, but if I was designing a dark website for example, I would set the background as a very dark grey rather than black, and I see most of the industry seems to follow this. The main background on this website for example could look white at first, but it's a light grey. Subtle changes like this can make a world of difference, so the best thing to do is just experiment - rather than just saying "this box should be blue" and slapping the first blue you see on it, experiment with different shades and see what comes out best.
If you wanted your site to be as accessible as possible, you can also check to see how they would look to someone who is colour blind using tools like this. Certain colour combinations may look fine to the average person, but could be impossible to read by someone with a certain type of colour blindness. (Note: I'm not colour blind so I guess I can't really say with certainty how accurate these services are).
As always and most importantly, your colour choices should accurately reflect the vibe and style you want your website/business to have.