Transferring WP domains from other hosts to Dreamhost using Transmit & a Mac

Hi Everyone. Yes another newbie & Mac user to test your patience with stupid questions !
I’ve been a Dreamhost client for a few months & have been making the effort to educate myself on setting up domains on Dreamhost & transferring domains from other hosts across to Dreamhost. I find the Wiki articles useful to a point although I do get bogged down with the technical jargon & find there’s not much Mac content & nothing on Transmit.

I’ve purchased a number of WP blogs over the last couple of months, all of which have been hosted by Hostgator.
I’ve had a couple transferred to Dreamhost by some of the kind vendors I’ve purchased the sites from but I feel that I can’t rely on the generosity of others & need to learn how to do it myself. Also most vendors are familiar with & can help with cPanel transfers but have no idea when it comes to Dreamhost.
Anyway I’ve purchased Transmit for this purpose & would appreciate it, if those of you who are familiar with using this software with the Mac could give me some tips on transferring files across. Even if its just pointing me towards articles on this subject it would be most helpful.

I’ve got a WP domain at the moment, I’ld like to transfer across from Hostgator. I’ve had the Go Daddy registration for the domain transferred across to me & have modified the nameservers for the domain pointing to Dreamhost. I’ve set up a domain of the same name on Dreamhost & downloaded the WP software. The standard “Hello world” front page now appears. I assume that the hosting for the domain is now on Dreamhost. However I now need to transfer all the old domain information (ie the WP files) across.

I’ve received the server details, username, password & port from the previous owner. I’ve been able to connect to the FTP server for both the old site on Hostgator & the new site on Dreamhost using Transmit. A series of WP files appear in the two windows of Transmit, many with the same name. Now I’m stuck. What’s my next step ? Or, have I got it all wrong or was there something else I should have done before I got to this stage ? Advice in simple language would be greatly appreciated. I also have another domain which I’ve received in a zip file via email from the previous owner. What’s the best method for setting this up on Dreamhost using Transmit ? Many thanks.