I'd recommend a "nifty 50".
Nearly all camera manufacturers offer an inexpensive, 50mm, 'fast' lens. Priced at less than $150, it offers a much wider aperture than your standard 'kit lens'. A wide aperture (1.4, 1.8, or 2.0) means you can get the subject in focus but the background will be nice blurry. Shallow depth of field (DOF)... that's a good thing when you're making portraits!
Canon 50mm f/1.8 II
Nikon 50mm f/1.8D AF Nikkor
(If these links are broken OR if you want a different brand just google "nifty 50" plus your preferred lens manufacturer such as canon, nikon, olympus, sony, tameron, etc.)
It's all about the eyes! At f/1.8 or f/2 it's really easy to miss the focus because the depth of field is sooooo shallow. It will take practice and you'll probably have some shots where the eyes are out of focus but the ears or nose is in focus. Keep practicing and you'll get the hang of it!