When I want to search for an image and I get the image results, I do not see a download option when I select a particular image (although the URL is of a .jpg file)
Is there an app for that or some way to do it with the existing options?
Google Image Search seems to be doing something funny with the way it shows you image previews.
What you need to do is when you run your image search you should see the grid of all the picture results:
Tap the picture that you're interested in and it should appear larger on a black background. When that first loads you should see something that looks a bit like a toolbar above the image. The toolbar disappears after a few seconds, just tap the image again to get it to reappear.
You should see a View Full Size "button" if you tap that the image appears actually full screen and now you can long-press on it and save it as normal.