Edit: written before I saw the version used in the tag has changed. Comments still apply, but since you are going to install access 2010, make sure you install the 32bit.
you have left it rather late to upgrade (about 10 years) it is simple, just open your file in your current version (2010 according to your tag) and select File>Save and Publish>Access 2002-3. You'll need to do this for both the front end and the back endIs it simple to upgrade to 2003? Do I just search Google and download it?
I note you are using access 64 bit. There is no benefit in 64bit since access makes no use of the additional functionality, If you can, reinstall as 32bit. To check your bit version go to File>Help
There are things that 32bit uses which are not available in 64bit so if you are using 64bit, I would not upgrade to .accdb just yet. .accdb uses a different database engine (ACE) from .mdb (JET). It is unlikely that this would cause a problem in the back end, but you may find some things stop working in the front end - all depends on the code your developer created.
There will come a point when 2003 will also no longer work with later versions of Access so personally, I would look to upgrade to .accdb in the not too distant future. Just don't do it until someone who knows what they are doing can take a look at your application and do the upgrade for you.