Writer: how to change the colour of a table cell?

I have a table. How do I make one cell red, or blue instead of plain?

There are two answers to your question. A cell may contain a single or several paragraphs. It the depends on whether you target the cell as a whole or some paragraph within it.

##Cell background

Put the cursor inside the cell and Table>Properties, Background tab. Here you can select Cell, Row or Table. Choose a mode Non, Color or Bitmap and the desired decoration.

##Paragraph background

Pur the cursor inside the paragraph and either modify the paragraph style, eventually creating a specific one to avoid colouring unwanted paragraphs, or Format>Paragraph (this is direct formatting and should be avoided). Got to Area tab and proceed as above.

Note that paragraph background does not cover the padding area defined in table properties Borders tab.

With paragraph background, you can have several colours in the same cell.

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Click the cell or select several (by sweeping them); Menu: Table>properties>background/Cell, click None/Colour.
Or click TableCellBackgroundColor in the Table toolbar