Can pilots go "offroading"? In other words, is there an airspace where you don't need a flight plan? Or maybe just a flight plan to go there and then "joyride?"
I am assuming from your question you are not a pilot.
So, as a general answer, in the U.S., flight at altitudes at and above 18,000 feet (5.49 km) above mean sea level requires a flight plan (an Instrument Flight Rules flight plan - known as IFR) and "joyriding/offroading" as you put it, is not permitted.
However, at lower altitudes (in controlled and uncontrolled airspace) a flight plan is not required in the vast majority of airspace in the U.S. and "joyriding/offroading" is permitted (there are certain Federal Aviation Regulations [FARs] that may apply, such as minimum altitudes or flying a correct altitude for your direction of flight, etc.).
Below 18,000 ft., there are some exceptions for certain types/categories of airspace areas and weather conditions where a flight plan is required, and/or "joyriding/offroading" would not be permitted, depending on the type of service you may or may not want (or require) from Air Traffic Control (ATC) and because of various FARs.
Also, there are many areas where a flight plan is not required (as mentioned above) but communication with, and sometimes authorization from, ATC is required.
Lastly, even in good weather when flying ("joyriding/offroading") it is a good idea to file a flight plan and communicate with ATC. Communication with ATC will help you avoid other air traffic and a flight plan will assist in locating you if you have to make an emergency landing or have an accident.
Note: I fly in EASA-land (EU) but I'm assuming the situation should be similar in the US and other countries
As @757toga described perfectly, there are situations where a pilot does not need a flight plan, mostly in low altitude, uncongested areas ("uncontrolled airspace"), in good weather at daytime. This varies country to country.
Equally importantly, however, you can go "joyriding" on a flight plan in most circumstances, sometimes even in controlled airspace.
In fact, 90% of the PPL practice flights in my area are on a flight plan, but they list some remark like "MANOUVER PRACTICE NEAR [village] BETWEEN [min-altitude] AND [max-altitude]". There's nothing wrong with that in uncontrolled airspace, it's actually usually much preferred than not filing a flight plan at all, for obvious reasons. *
In many situations, you may even be able to pull this off inside controlled airspace, if the controller permits you. If you'd like to do that, you can either just call them on the phone in advance, or you may need to get a written permit, depending on the place. But if traffic and other factors allow, the controller can allocate a block of airspace for you, inside which you can pretty much "do what you want". This is also how some special operations are done like photography, airshows, etc.
* Advantages of having a flight plan even when not required include:
- Lets controllers and thereby other traffic know about your intentions, which helps you to see and avoid eachother.
- In case you get into an accident or simply "disappear" from radar, the flight plan lets the authorities know about your existence so they can notify search and rescue services. They also know your approximate intentions, and important information like # of people and fuel on board.
- In my country in addition to the above if you don't have a flight plan and you require search and rescue for any reason, you are financially liable for those services.
- If you accidentally get into bad weather and would like an IFR clearance to penetrate clouds you can't get one without a flight plan (at least here under EASA rules). You can technically file a flight plan from the air, but that's one extra step in an already time-critical phase.
To be a little less technical, and simply answer the specific question, yes, there is a ton of airspace in the USA where you can fly without filing a flight plan, like almost all of it under 18,000'. I've logged all 800+ hours of my flight time without filing a flight plan.