Hi
To change the domain name you will need the nameservers from vistaprint and change these in the DNS settings in you 1&1 account, it couldtake approx 48hrs for it to work, you will be able to keep the same emails you will just have to set up the smtp and pop3 in your email client to run with vistaprints servers, judging by the packages vista offer, speaking from experience why would you want to move to vista print?
For a little more money a month you could have a bespoke website made by a professional in this industry and the site will be constantly updated and altered to run with the ever changing rules of google. Giving you what you really want from a website and not just lost in the billions of pages already out their
If you do want just a box standard website for customers to write your web address and you not worry to much about geting in the first 8 pages of yahoogle then these sites are great, if you want your website to help obtain more customers and work for you then Im afraid to say you need a little bit more than what vistaprint is offering.
Hope this helps