Updating site and site migration Advice

Hi All
My wife is a graphic designer and is re-designing (new site will be converted to Wordpress theme) an existing site for a client. The client is also going to be moving hosting at the same time.
The client gets about 60% of business through his site and is ranked number 1-5 in Google for several keywords, so I am using a 3rd party company to do the site migration, as I don't yet feel confident enough to do it myself.
It is essential that the site is migrated properly so as not to lose client rankings, so I want to verify 3rd party company will be doing all necessary steps i.e. 301 redirects, broken links etc.
I would like to reach out to experienced Warriors who could advice on all steps that need to be taken to make sure clients rankings are not affected when moving to a new site and migrating to new hosting. Any advice would be really appreciated.
Also what is the best way to work out what keywords site and pages are ranking in Google, client has built current site himself and doesn't have any sort of keyword list. I will need to work this out to make sure content on new site pages is optomized for relavant keyword.

Thanks in advance for responces.

Regards
Joner