To future readers: Note that this answer was written in June 2013, when Wheezy (Debian 7, released in May 2013) was "stable" and Jessie (then-upcoming Debian 8) was quite new as "testing". Consider the impact that any further development may have on the validity of the answer in your particular situation before you proceed with an upgrade.
To ease the transition, I would recommend that you do this as a two-step process. Or three: make a backup first. The probability of data loss during a Debian system upgrade is small, but non-zero, and configuration files very well may get overwritten or rewritten as part of the upgrade, and there is always the possibility that the translation isn't perfect. It should all be perfectly possible to do without having to reinstall from scratch, however.
First, follow the normal upgrade path to the most recent revision of the stable release, currently Wheezy (the link leads to the i386 upgrade manual; substitute for your architecture as needed). This basically boils down to editing
/etc/apt/sources.list* to name
wheezy rather than
squeeze and then doing
apt-get update followed by
apt-get -u dist-upgrade, but read the documentation for the gory details before you proceed. I recommend test-driving Wheezy for a bit, to make sure nothing is obviously broken. If it's broken in Wheezy, it almost certainly won't magically heal in Jessie or Sid.
When you feel confident that the system is working properly under Wheezy, edit your sources.list files again, this time substituting
wheezy. From the Debian releases page (my emphasis):
The "testing" distribution contains packages that haven't been accepted into a "stable" release yet, but they are in the queue for that. The main advantage of using this distribution is that it has more recent versions of software. /.../ The current "testing" distribution is jessie.
The "unstable" distribution is where active development of Debian occurs. Generally, this distribution is run by developers and those who like to live on the edge. The "unstable" distribution is called sid.
If you want bleeding edge, you want
unstable. Note that Sid is prone to heavy breakage. If you want more recent versions of packages but semi-stable, then you want
apt-get update followed by
apt-get -u dist-upgrade. At the time of this writing, there does not seem to exist a specific installation guide for Jessie/testing, but it is still soon enough after the Wheezy release that the differences should be relatively small and a simple dist-upgrade should suffice in the majority of cases. The
-u will make apt-get prompt for confirmation before actually performing the upgrade, giving you a reasonable chance to review the proposed changes before committing to them. I recommend not removing that command-line option unless you are feeling truly adventurous (and have really good backups).
Oh, and in case I forgot; make sure you have good backups of your data before you begin. As with any system upgrade, I strongly recommend doing it directly from the terminal (not through an X session, and certainly not through an X session without a terminal multiplexer such as
tmux) to avoid problems when services get restarted during the upgrade process.