What airspace are ultralight pilots allowed to use?
Ultralights are regulated under 14 CFR 103. There are very few regulations that apply to them, but there are specific rules on airspace in 103.17:


  §103.17   Operations in certain airspace.  
  
  No person may operate an ultralight vehicle within Class A, Class B,
  Class C, or Class D airspace or within the lateral boundaries of the
  surface area of Class E airspace designated for an airport unless that
  person has prior authorization from the ATC facility having
  jurisdiction over that airspace.


That means, you can fly in (most) class E airspace and all class G airspace. You can find more information about those airspace classes in this question.

As for your other questions:


A flight plan isn't required even for aircraft, and ultralights aren't subject to the regulations that would require them anyway
Ultralights aren't required to have radios or talk to anyone; you usually fly so low that you aren't in anyone's way
It's not at all the same as flying a Cessna; in fact, you don't even need a pilot's license to fly an ultralight
See this question for information about launching from national parks; state parks would be subject to state law instead but the principles are probably the same


You might want to look through all our ultralight questions for more information.