Good deal! From my experience with clients in similar situations, I suggest that the main training you will need is regarding three main subjects:
1) The use and operation of your chosen CMS - Both the Joomla! and WordPress applications have extensive user communities on the web, and there are many "free" web-based and documentary tutorials on their use available on the web. Joomla! is the more complete and powerful of the two, though it is somewhat harder to "get your head around". You should be able to get yourself up to speed on either or both of these applications using these resources.
2) Preparation of text and graphics for use on the web. The re-sizing and optimization of graphics on the web is very important, as most modern digital cameras produce images far too large (in both dimensions and filesize) for web use. There are many tutorials on the web for preparing web graphics, and you could avail yourself of these to resolve that issue. The main issue you will face with text is the understanding that MS Word is worse than useless for preparing text for the web, as MS mangles text cut'n'pasted into most CMS systems with MS specific codes - this can be avoided by using a more generic editor, or by using the CMS system's "cleaning" facilities to re-purpose such text. Again, the web has lots of info on this process, as do the various CMS user communities.
3) Management of the site's administration, to include timely backups and management of email accounts. The Dreamhost wiki, these forums, and the user communities for the various CMS applications have lot's of help available for these functions.
All that said, the remaining consideration I would advise you to look into is the careful organization or content for your site. Ideally, this should be done with the CMS you will be using in mind, as the methods for organizing and presenting data in a logical structure varies somewhat from system to system. Generally speaking, I feel that "simpler is better", as it makes your site easier to manage and information will be easier for your visitors to find on the site. This is another area in which you may want to seek out a professional, though again, there are many educational resources that deal with this readily available on the web (and in the various CMS user communities).
Finally, many organizations in your circumstances choose to approach the process in a two-stage manner: first they develop their "new" site, and test it, in a "staging" environment while continuing to operate their "old" site (or move the old site to the new server and develop the new site in a staging area on the new server) and then, when the site is ready to go, launch the new site in place of the old site.
This allows the organization to maintain an active web presence while updating to the CMS. Fortunately, DH allows you to host multiple sites at the same time (multiple domains, sub-domains - like dev.yourdomain.org, etc.) at no additional cost, making this process not only feasible but affordable (it's hard to beat "free!").