I'm not a user of Premiere, but here are a couple suggestions to help you figure this out.
First of all check out the web sites at Doom9 and VideoHelp (just Google for them) as they both are excellent sources of information, help, and tutorials on video recording issues.
That said, yes, Nero will do the basic transitions that you want to do.
The other point is that your are converting avi to mpg prior to you edits. I would just use the native format the file is in and make your edits in that format first, and then reencode it to the final format you want. It seems like you are doing extra work and simple is always better.
Lastly, the codecs you have installed on your computer may be a problem if you have several video editing programs installed. While different codecs can co-exist on the same machine (some can't), some programs hijack the file associations when they are installed. When this happens programs can use the wrong codec instead of the one for which they were originally designed. (The order of the codecs listed in your associations can influence which one is used first)
Hope this helps.