Unfortunately, the walk to Ellis Island probably won't be enjoyable or scenic. I would actually plan to go to Ellis Island some other time. I just had their pancake special for breakfast a bit more than a week ago, and it was quite good.
Since you are interested in the fountains, what I would do is exit Bellagio out the main entrance. The bell desk will be on your right. Continue past the bell desk, continuing to your right, and you will find an escalator up to the moving walkways. Go up and take the moving walkways to the strip. You can stop between the moving walkways, and also at their end, to enjoy the fountains.
You then take an escalator down to the strip, and head directly north on the strip, in front of the fountains, and view the fountains from that sidewalk. Unfortunately, that is a popular stroll and you may be heckled by street preacher types who know there will be people there that they can heckle.
After progressing north to Flamingo Blvd., you can cross over to Bally's on the pedestrian bridge. Then you can continue by walking through Bally's, down to the Bally's sportsbook, and exit onto Flamingo there. If you'd rather not walk through Bally's, you can take the other pedestrian bridge over to Bill's and then walk east on Flamingo, past the Westin, and then south on Koval.
I find walking through Bally's, or walking along the north side of Flamingo (from Bill's) to be much more enjoyable than walking along the south side of Flamingo from the pedestrian bridges.
