I want to test my new cruiser and kill some basic pirates, what's the quickest to fly out and get into a fight with some Non Player Characters(NPCs)?
Find an agent that gives combat missions. You should go with lvl 1 or 2 agents. http://eve-agents.com/
For very basic pirates (rats), and assuming that as you're a new character you're currently in high sec (probably 1.0) space.
The easiest way to be sure of finding them without actually accepting a mission from an agent is probably launch your ship out of your current station, bring up the system map and look for asteroid fields. Fly to the nearest asteroid field, stay there for a few seconds to see if any pirates spawn, if not fly to the next asteroid field and look there. You're bound to find some rats in one of the in-system asteroid fields and they're likely to be pretty low level, the sort of ones that you can take on without too much danger.