Anytime except summer! New Orleans is surrounded by water, and is barely above sea level at best. The humidity is high year-round, making summer activities difficult to endure. Winter in southern Louisiana is usually perfect - clear skies, bugs have died down, and highs in the 70's.
What to do if you have a sunny Saturday?:
Start with a run around Audubon Park in Uptown between St. Charles Avenue and the Mississippi River. The road inside the park is 1.7 miles long, and is open for running and cycling without cars. To extend your run, head down the median strip of St. Charles where the street cars run (called neutral ground, locally) for a few miles and take a street car back to Audubon Park.
From Audubon Park, you can also hop on the levee along the Mississippi River, which goes all the way to Baton Rouge. That levee is the best off-road cycling in the city, although there will be lots of pedestrians on the weekends in the first few miles.
Where can I go to get out away from the city, without driving too far?:
The north shore of Lake Pontchartrain is the best place to get away from New Orleans. If you have a car, take the causeway across Lake Pontchartrain.
Fontainebleu State Park and the Tammany Trace bike trail on the north side are classic southern piney woods country. Abita Springs makes for a great getaway spot as well.
What are some great weekend getaways?:
The Mississippi Gulf Coast beaches are a classic New Orleans getaway. The oil spill may tarnish those beaches, so for inland fun, head to Alexandria for the Azalea Recreation Trail, or up to Natchez, Mississippi for the beginning of the Natchez Trace Parkway, a 400 mile scenic drive to Nashville Tennessee that is a favorite of road cyclists.