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.