How do I completely ignore styles in Calc? I don’t want to use them, don’t want to know about them, don’t care about them. I want to click on a cell then click on the Font Color icon and have it actually take effect.
Congratulations! You are ignoring them already 
You might also be ignoring conditional formatting and Format > Cells, neither of which are Styles. Also you might have View > Value Formatting on. You are welcome to upload a sample by clicking the up-pointing arrow in the edit window.
In addition to Help, the Calc Guide is a useful reference, free download from English documentation | LibreOffice Documentation - LibreOffice User Guides
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You can not ignore all of them. At least one Cell Style (the Default Cell Style), and one Page Style (the default one) will be applied when you create an empty new spreadsheet document based on the Default template. (And there are more predefined Cell styles what you can apply withuot creating new styles. You can not delete the predefined styles from a document.)
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The most valuable formatting feature of the LibreOffice is the Styles. I suggest you to use them.
The manual (direct) formatting is not an efficient method to format any type of the documents. You can do it, but that is not the best way.