I sometimes use Illustrator as a drawing board for online lessons, and I'd like to have a good calm, but vivid full color palette for writing or drawing, preferably CMYK. Illustrator has a lot of built-in color palettes, and the best one for my use is just Print Brights, but it only has 6 colors, lacking light blue, pink shades, etc. Where can I get good palette like this with 10+ or so colors? I could've built it myself, but I'm not very good with colors specifically
Make one yourself. You don't have to be "good at colours", although not being colour blind would be an advantage.
You could find a rainbow gradient by simply searching for one online in google images. Here's one you could use
Then in Illustrator draw a rectangle, and with the eyedropper tool sample 10 colours, adding each as a new swatch as you progress, obviously not forgetting to convert each one to CMYK mode first. Note that CMYK colours (particularly bright blues) will not be as bright as an RGB bright blue.
Afterwards you could arrange them in a folder to organise them.