When you bring into Illustrator an image from the internet you need to break it up into its different components so that you can change its colours.

Select the image

Window > Image Trace to open the dialogue box

Judge how many colours there are 128 in this case and change as appropriate

Click Trace - if your image goes black then press Ctrl Z  and change the mode to colour and then click Trace

This is another way to reach the same goal

Choose Object > Image Trace > Make and Expand

Your image then is broken up into smaller elements

You can now play with the colours and experiment

Go to the colour guide

Choose a colour swatch, click on the colour wheel to open the panel

Click on each colour in the edit window at the top and then at the bottom double click the colour and then choose another colour and continue til the end and then click OK.

If you want to change individual colours then click on the direct select tool and use the colour swatches.

Hold shift to select several areas at the same time

Choose a photo and open in Illustrator

Window > Image Trace 

in the dialogue box choose colour and then 30 colours

Object>Image trace>Expand

Direct select the whole image and go to the colour guide

image Trace - colour - 8 colours

You could choose your image and apply the image trace as already shown and rather than using the colour swatches you can live paint and choose individual aspects to colour.

Select the object with the direct select tool and then on the tool bar choose the iive paint option.

Choose your colour from the colour swatch in the tool bar and start painting selected areas.