Printing both black and coloured text and over coloured panels - how to set it up correctly

My home printer is printing all the text as just black so I am a little concerned.

Do you have Text as Black unchecked?

I would like to know if it is okay to set the black text to overprint while leaving all the coloured text as it is?

Yes you should overprint black text and the [Black] swatch is set to overprint by default in Preference. You should not overprint colored text, or at least make sure you keep Overprint Preview turned on so you see the affect of any overprints. By default objects filled and stroked with a color other than [Black] are set to knockout.

For colors that knockout, it is the printer's responsibility to correctly handle the trapping. InDesign does have a trapping function, but the traps are applied at output and not to the objects themselves, so there's really nothing you can do to control how a page is trapped downstream.

Depending on your preferences, it's most likely that the black swatch at 100% will automatically overprint. (See “Appearance of Black” preference—the overprint button should be checked.)

How have you set up your color swatches? Depending on the way your finished document will print, you should check to see if they are set as spot or process. Even if they are set as spot, if you are writing to a .pdf and find out the client wants them process separated, there is a checkbox for making all spots process in the "Output" section of the .pdf output dialog.

There's two parts to this.

1) Setting up your InDesign document correctly. I suggest you take the introductory course in this online video tutorial. The first chapter is called "Learn InDesign in 30 Minutes". You can perhaps take the rest of this (or other InDesign courses subsequently. You can get a 30-day free trial. InDesign CC 2018 Essential Training

2) Don't print your InDesign directly from the application but create a PDF and print from that. Probably the best choice of type of PDF (there are several) is to use the InDesign Acrobat PDF Presets and select (for a desk top inkjet printer) High Quality Print.

By the way work in the RGB color mode and let your PDF do the conversation to a suitable space.

I suggest you use layers.

One for the black text and one for the coloured text.

To be safe, supply to PDFs, one with the black text layer, and the second with the coloured text layer.

This will make it easier for you to proof on your printer as well, especially if it is a non-Postscript device.

On the text on coloured backgrounds, Paper will work great, so you're OK there.

HTH