Mariner,
The best advice I would give to a non-techie is (as you said) that if your computer is under warranty, call customer support and let them figure out the problem and/or step you through getting it fixed. Dell is very good at step-by-step instructions.
Re: your advice to charlie brown, I agree that a router is an absolute must, even with the windows firewall or third-party firewall activated. A known issue with prior versions of ZoneAlarm was that after periods of inactivity, sometimes the computer would need to be restarted in order to connect. Even with a router and DHCP enabled, usually a new IP is not assigned, unless another device grabbed the old address in the meantime. PINGing the router is a way of checking where the disconnect is.
If you are buying memory or other hardware I suggest the following sites:
(for daily bargains)
(for reasonable prices. Newegg has great prices too, and is more of a geek store.)