Is my computer operating at a typical temperature?
While I cannot speak from experience on that, personally I find that, under normal circumstances, you should only investigate your CPU temperature when you already have problems such as instability. All modern processors have thermal protection circuitry that prevents them from operating at temperatures that would damage it at it's standard operating settings (voltage and clock speed), so it is unlikely that you will be able to use a system and have it "take damage" from operating hotter than normal.
and at what temp should I begin to worry?
For me, when I get or build a new system, I'll usually make a mental note of what the "standard" temperatures seem to be for the various components. This gives me something to compare against.
In your case, the 60 degrees is in fact under load, and is well within the specifications for the processor. My suggestion would be to use the laptop normally, and get a idea of it's normal operating temperature. If it get's excessively close to the thermal limits- which I believe others have cited as 70 degrees or 80 degrees- you could consider returning it under warranty.