Risks involved in upgrading to SQL Server 2008 R2
That's a really big question so let's break it up a bit.

What can I do in advance?

Start with some required reading. 


2008 R2 Backwards Compatibility
2008 Backwards Compatibility


These links have links to further information such as


Deprecated SQL Server Features
Discontinued SQL Server Features
Breaking Changes
Behavior Changes to SQL Server Features


Read each of them to see what major stuff is changing. Pay particular attention to features you are using.

In addition you should use the Upgrade Advisor.  It checks for installed components and identifies those you will need to fix either before or after the install.


In-Place vs Side-by-side

Lots of pro's and con's here on both sides.

In-Place

Pros


MUCH easier. All of your configurations stay the same for example. Also the connection strings for your applications probably won't need to be changed.
Cheaper. No second set of hardware required.


Cons


Backout is difficult to impossible.  If something goes wrong you are going to have to power through and finish because backout involves creating a whole new server and reinstalling SQL then restoring backups of your tables.


Side-by-side

Basically the pros &amp; cons are the opposite of In place.  

Pros


Safer - If something goes wrong you kill the new version and just continue on with the old one.  Then you can try again later.


Cons


It's more expensive because you have to create a new set of instances probably on new servers.
It's more difficult because you have to change connection strings, make sure all of your configurations are the same, etc.


Now you can mitigate the expense of the side-by-side by creating a new instance on the same server, moving everything to it, then uninstalling the old instance. It works and depending on your situation might be the best idea.


General Risk

Honestly the move from 2005 - 2008 R2 isn't that bad.  It's nothing compared to 2000 - 2005 or 2008 R2 - 2012 (mostly SSIS changes).  I'd say with careful planning and reading you should be in good shape.