Better than 50/50 chance that your old chain was, indeed, worn, and that the gears your talking about were worn down in the process.
Examine the teeth on the cogs that skip. Do they look identical to the teeth on the cogs that don't?? If not, you probably need a new cassette.
Most of the pictures on this page are of "healthy teeth" on a cassette:
http://www.parktool.com/repair_help/FAQcogs.shtml
It's also possible that you have a rear derailleur that needs the cable tension adjusted.
The derailleur adjustments can be found at:
http://www.parktool.com/repair_help/FAQrindx.shtml
The Local Bike Shop should be able to take a quick look, if you can't get through this, or if it doesn't help.
Write back if you get any further information from these steps. Maybe we can offer more advice.