Let's say I created a suitable color palette. What would be the best way to make sure the colors are used consistently? – Giving it names like primary_700 or accent_1_700 seems a bit too technical.
Use a css preprocessor, and define variables with the technical names but classes that end up in html should have descriptive names. Something like this:
This makes your color palette defined explicitly and easy to change in one place, and makes which color on the palette you're using in a given context defined explicitly as well.