InDesign changes the colors that I choose

Pretty much any color I choose it will automatically change it.  

InDesign’s Color Picker is context sensitive. The color you pick depends on where your cursor is in the dialog. You can see it in @Brad @ Roaring Mouse’s capture where he’s getting Add CMYK Swatch when his cursor is in the C field and an Add RGB Swatch when the cursor is in the RGB R field. Hex is an RGB notation, so to choose #00FF00 your cursor has to be in one of the RGB fields with Add RGB Swatch showing:

When placing images make sure to embed a color profile when you save them.

If your end goal is to print this document, you have to remember RGB/HEX colours are only for the screen. They do not translate to a print space. Normally, InDesign assumes this is a document for print, so whatever you type into the HEX box may be converted to the CMYK values at right (The clue is it will say "Add CMYK swatch" above). This colour is the closest you will expect to be able to print based on your Color Settings, which truthfully, is a good thing for you to know. However, if you click in any of the RGB inputs, you will see "add RGB swatch". This will give you the color you want for your object on the screen, but you still won't be able to print it.

Some colours in the RGB space are easier to match in print, so, yes, depending on where you click in the color picker, there will be more/less difference.

You can then turn Proof Colors on or off (under View) to show the difference between RGB and whatever CMYK space you have set (Mine is GRACOL; yours may be U.S. Web Coated or something similar)

You may want to check your color settings — Edit>Color Settings menu command — which drives InDesign (and other Creative Cloud applications) to simulate output reproduction onscreen. Default settings for InDesign is to simulate printing on coated paper stock, but there are others which simulate other print and online/digital reproduction models. Color swatches are altered to simulate how the color will appear when viewed in the final product.

Think of it like taking a canoe across a river, where you have to paddle upstream if you don't want the current to carry you past the spot you want to reach straight across the river.

You may find that you need to define a custom color setting that does not cause alteration of the swatches. You can read more about InDesign color settings and how you can alter them to fit your reproduction needs by following this link.

Hope this helps,

Randy

One thing to keep in mind: RGB colors are approximate if you are trying to pull a color from a placed image. You ought to know the colors you want to use in your document and stick to one colormode.