I recently bought a new laptop (hp 2000 notebook, if it helps). I've had my iPhone for a few years now. I have downloaded iTunes, but when I plug in my phone to my laptop, nothing happens other than a message saying the phone is not recognized. I'm severely lacking in any IT skills. Am I skipping a step, or is something up with either of the two?

It sounds like your missing a driver for your phone. Once installed it should recognize your phone. All devices hooked to each other need a common way to communicate. The driver is that link that accomplishes this for you.

Have you been able to connect the iPhone on anouther PC? Also try a different cable.