I've just migrated a whole bunch of domains to these guys - siteground.co.uk and it all went relatively smoothly. Their on-line chat help facility is a life saver, particularly when you're unfamiliar with their systems.
Admittedly, the migration I did was not as complex as yours, but it did involve some DNS record changes to interlink other facilities into the sites.
The way I did it, was to attack one domain at a time (after all, it doesn't matter where your domain is served from) and transfer all the files relevant to that domain over. Incidentally, providing you choose the right package, you can have multiple domains on a single instance, all in their own root directories. Then create any sub-domains as necessary and finally make any DNS changes that were required. Remember, at this point, nothing is live from the new server. Once I was satisfied that (as far as I could tell) everything was ready to go, I initiated the change of DNS servers to siteground, which already had all the relevant information on them.
In the main, it all worked very well and really the only glitches were of my own making, where I failed to think through possible scenarios. But these were quickly dealt with either by me, or with the help of the siteground guys.
It's not as daunting as you may think.