Joomla/database expert to speed up website

Since transferring our website to Joomla 1.5.21 from a basic HTML site in Oct 2010 the home page load speed has increased to at least 4 secs which has negatively impacted google and alexa rankings.
The original HTML site was fast and achieved quite high alexa and google rankings and was on the same server as now (shared hosting from godaddy).
This speed problem could be from one or a number of factors:
Database
Template issues
Server (godaddy shared hosting) issues
Content
Other

From my limited experience I think these are possible solutions?
Optimise database or transfer to new database
Transfer to new/different type server/provider
Optimise template code


It is vital that there is no disruption to the availability of the site to visitors and that no changes are made that will have a negative impact on serps/seo.

I require someone who has experience in joomla backend, php, database,seo, to try and speed up the site to the optimum for the current site. It may be the case that the site, as is, can't be sped up but I would like a report on the options available.