The Dell web site indicates there are several variations on your computer, such
as screens from 800x600 to 1400x1050, plus different CPUs and memory. If you
were to describe it more fully you might get better advice. (CPU, memory,
screen size in pixels.) At minimum, it's less powerful than today's netbooks.

You don't need a wireless router to see if Ubuntu can talk to your wireless
adapter. If Network Manager can see the adapter, it might tell you there are no
wireless networks available. If you are in a city, it will more likely ask you
to choose among the half dozen networks it can see! My son lives in an
apartment building, and his wireless adapter can see a dozen networks.