Basically, all you need to do is download your sql tables. Look for something mentioning Export tables, or databases. I'm not qualified to go deep into SQL management, refer to the help files of your particular script or sql access program. Hopefully other users here will step up to the plate as well.
In terms of the nameing convention, you can use a different DB name; that doesn't matter, but the tables inside should all look the same for the most part, unless you specified something other then xoops_ as your table names prefix.
xoops_users
xoops_config
etc...
Install your XOOPS on the new site (domain), log in as admin and install the forum. Get that running first. Once complete, we can help you try and get your users over. Once thats complete we can get your posts over as well.
Keep your site going as usual, they don't need to know whats going on unless you tell them. It doesn't matter if you get new users, or if they make new posts. Once you understand how to import and export current tables, you can close your old site anytime and move to the new server with current information in the database.
I would do it for you, but I don't like accessing other people's accounts, as I would need full access to both your old and new account, full complete access to both domains in order to achieve the results you need. Something I will NOT do, so please don't ask. I've been asked before, but I can not and will not be responsile for anything that happens so I play it safe and say no when asked. Sorry. Others here may do so, but it's at your discretion.
Essentially, you can do it yourself, you have the head for it, just lack the experience. I'd be willing to walk you through it step by step as much as I can either here in this post, Private message, or prefeably through MSN when I have the time for live support.
How you go about your migration is up to you, but we'll be here to help all the way. It can be done