Aside from losing the link locations the new flickr (which as far as I know is still in testing you can still revert to old flickr) I think the upgrade has overall improved the interface - preview picks are bigger and more of the screen is used.
For sharing pictures there are 2 methods with the new version:
HTML Code
1) click on the "share this" drop down menu
2) from the list that appears select the "Grab the HTML" line.
3) Select the size that you want to share from the little dropdown menu
4) copy and paste the HTML code that is shown
URL
1) click on the "Actions" drop down menu
2) go about half way down the menu to "View all sizes"
3) in the new page click in the image size that you want the url for
4) once the image has loaded right click on it and go to properties/image properties (different browsers will have slightly different wordings).
5) in the little window that opens copy the "Address" line - that is your direct url for that size of this photo.
Note some browsers will have a "Grab/copy" url command when you right click so you can use that if you prefer.
Note 2 the URL link in the share this drop down does not link to the photo alone but to the photo page itself. So if you want people to view the page in your flickr you can use that link; whilst if you want to embedd the image into the forums (or a blog) or direct link people to the image alone use the url method outlined above.