"a blowout just airing the tires up, twice," - Go slowly, not pinching the tube.
"rear derailleur" - Old bike? This is why i didn't buy used. Have it inspected by a shop before riding.
"hit some uneven pavement going 15mph " - Lighting/vigilance/practice and using your legs/arms to ride (looks like you needed The Art of Cycling before getting started)

Have a route that avoids heavy traffic and bad roads.