It's great to be able to grab some random colors for inspiration. All too often it's easy to just go with the same old color combinations that we tend to gravitate to naturally. To avoid falling into that rut, it's good to switch up the colors. Some situations where a bunch of colors are used at once are:
Often there are many practical considerations; we can't just use any color. For example, a pale yellow on a white background is not going to work for graphing purposes. Or if we have two colors that clash, it's not going to look good in website design or graphics design.
If you're looking to generate a set of colors that work well together, there are several options.
Paletton is an intuitive tool that suggests palettes and, more importantly, shows how these colors would look next to each other in a simulated website layout. This is an ideal tool for web designers.
My favorite though is Coolors because of how modern the colors are. You can see that the palettes they generate would look fantastic in a modern website. Both of these options give you the hex codes so that you can use the exact same color in your graphics software or web design code. If you have an account with them you can get extra features, but anyone can generate and use palettes for free without being signed in.
However, neither of these are great when it comes to simply wishing to add extra colors to your existing stylesheet, as neither of them give CSS code for it. You'd have to add it yourself, which is no big deal since they give you the hex code, but it's still a little frustrating.
If instead of a palette you're looking to simply generate a bunch of colors that you can add to your stylesheet as individual colors, you're in luck. Color-hex.com is a wonderful informational source for each color. Although it doesn't generate random colors per se, its home page shows the most popular recent colors, which is somewhat close to having a random color generator. When you click on a color, it shows you lots of information, including hex code and CSS. You have to scroll down almost to the bottom to get the CSS though.
An alternative is the random color generator at JotSplash.com. Besides giving you the hex code, it also gives you CSS that you can literally copy and paste into your stylesheet to start using that color. It generates several random colors at once, and the hex codes and CSS are available right from the home page, no clicking or scrolling necessary.
So if you're doing web design, those are great options for you.
If you're seeking color inspiration, whether as palettes or individual colors, there are several options available. Depending on what you're seeking to do, one or another of these options above will suit you better.