No, downloading to your PC is only the first step of the upgrade.
If you remember how it was when you first installed ZenCart, the process is almost identical when you upgrade except you won't have to add products, prices, and all that hard work. Those will carry over automatically. Most of the configurations you made originally should also carry over, except for shipping and payment modules (you need to write down some of the info in admin for the payment and shipping modules and manually re-enter after the upgrade. All that is clearly explained during the upgrade installation process.)
It takes an expert perhaps 10-15 minutes to upgrade, a little longer if it's been awhile since you had to work through an installation or upgrade.
If you have not tweaked your core files (i.e. the original installation files) then the upgrade should go very smoothly. However, if you have edited some of the files (other than those in a third party or custom template) then you may lose some of those customizations.
As a previous poster said, there shouldn't be any issues with templates, however there is at least one template I have encountered which needed to be tweaked for v1.3.8a. (a template named icAcademe).
BTW, if you have tweaked your core files, and you don't have any problems with your current install, it might be better to leave well enough alone and stay with v1.3.7. I have two websites, one with v1.3.7 and another with v.1.3.8a and see no compelling reason to rush an update on the older one. But there is a reason I am holding off...... I did modify some of the core files before I learned how the "Override" system worked.
If you have never tweaked your core files I say keep up with all the latest upgrades as most versions are well vetted before being posted. And, if at any future time you feel you need to modify Zencart files for your special needs then be sure to learn and use the override method. That way you can always upgrade without losing those customizations.
Ron
