If you were convicted of reckeless driving in Virginia, you will need to report it on the N-400 as a crime (class 1 misdemeanor offense). Speeding charges, however, are merely traffic infractions and likely have no need to be reported unless such is clearly being asked for on the application.
None of these, however, should have any adverse impact nor effect on your application for citizenship.
Don't mess around with immigration issues. Get an attorney asap! I've seen too many cases here in Florida, where I practice, that there are conflicting issues with various offenses and until you hire an expert in immigration, it's tough to navigate these issues alone.
In our experience, "reckless driving" charges are typically reduced in the case of a first offense. My concern for you, is that if you have a conviction, then I would be interested to see the rest of your driving history. You must report all charges, citations, fines, traffic stops and any encounter with the police on your N-400. Unlike the Adjustment of Status form, the N400 requires you to disclose all encounters with police and does not exclude traffic stops.