Browsers are very much a personal choice, there is no 'best' browser. I.E., however is no longer supported for mac so I wouldn't recommend that. If the feature you describe is a must, most Mac browsers allow simple dragging and dropping of images which will save them to wherever they are dragged to.
Firefox is a good browser, some say the best out there (but not me). It offers many add-ons to customise it, perhaps the most of any browser
http://www.mozilla.com/en-US/firefox/
Personally, I use Camino (
http://www.mozilla.org/projects/camino/) It is like Firefox but it uses a much more Mac-like User Interface.
I will sometimes use Safari, though it has its problems with some webpages in my experience. It does handle RSS feed very well though
Shiira is another browser out there. It is based on the same engine as Safari and has features such as a sidebar to view favourites and downloads and a page dock - like tabs with a preview of the page.
http://shiira.jp/en.php
Flock is described as a 'Web 2.0 browser' and includes blogging support, uploading directly to Photobucket and Flickr and a 'Shelf' to put content on that you want to come back to later.
http://www.flock.com/
Others include Opera (notable features include built in Bittorrent client), iCab (able to run on very low-specced macs), Omniweb (described as 'Safari done right')
Check out
http://darrel.knutson.com/mac/www/browsers.html
which has a pretty comprehensive list of what is available. Download a few and see what you like
