I have just done the same and it can be a bit terrifying if you have never done it before... so may I offer a bit of advice from a newbie at database transfering.
1: If you intend to transfer from one host to another try and keep the original one open until the new one is fully up and running. That way you can keep going back and forth when the mistakes happen (and they will!)
2: Trim your forum down as much as you can because restoring a big database to your new host can easily fail...many times!
3. I tried to reduce the size of my database via downloading as a GZip compressed file but my host kept coming up with errors all the time so I gave up on that..... However, I have only just seen that in the Config there is the option to enable GZip files.... I had this option disabled so I presume that may have had something to do with it (Can anyone confirm that please?)
4. Even though I reduced my database down from 20mb to just 5mb I still used bigdump which I found very easy and you can watch the progress as well.
If you havent done so already I advise you to read all about bigdump and the really helpfull auto cookies http://www.phpbb.com/kb/article.php?article_id=374
Hope this is helpful......