You should be able to upload the pics to R2R from your computer and not use Photobucket. Not sure if new members have that capability though. Look to the lower right hand corner of the post pane and see if you see "Upload a File" button. If so, click on that and browse to find the image on your computer and just upload it. It will give you the option of using a thumbnail or post the full size image.
If you prefer to use Photobucket, I just use the Direct Image Link and post the URL in the window that opens up when you click the Image button in the upper tool bar in the posting pane. Like so...