Use the Insert menu for images. Click 'Images' to select from your Drive/Photos. Use Upload to select from your computer. You can upload only images. You can also double-click to get the "Compass Menu".
Only JPG, PNG and GIF images are supported.

Google Photos Animation

This involves using the animation feature in Google Photos. The procedure is explained by Peggy K, a top contributor in the Blogger forum. You can find her post about this in Google+. She explains about using it in Blogger but as you can see it also works in Google Sites.  You should view her detailed instructions but here is my condensed version:

1. In Google Photos click utilities in the left menu.

2. Select Animation

3. Click to select each image you want to include

4. Click the Create button

5. Click the more actions icon (3 dots) and add the animation to a shared album

6. Right-click the animation and copy the image URL

7. In your site click Insert, select Image and then Web address (URL)

8. Insert the URL you copied in step 6

To make a slower animation you can use an editor like LunaPic.  You can upload your fast GIF, change the speed, save the image and upload it to Google Photos, get its URL and insert it into your page.  (Don't forget to move it into a shared album)

Example of a cropped, zoomed in, image.

Before cropping it I captured the URL of the image. Then after cropping it I added a link to the full-sized image. The result is that the viewer can click the image to see the big image.  To make a link, click the image and select the link icon (looks like a chain link).

New Sites does not automatically make a link to the original image, thus requiring the above method to be used. You can send feedback to Google to request this feature.

See the video here for cropping and zooming  instructions