Reks I would replace the cart, just so you know your vinyl is seeing a fresh stylus, worn stylii can damage stuff. Then I'd hold off on upgrading anything else to see if vinyl really is for you.
A few hundred is a pretty decent budget. I would personally avoid any Grado, they do some weird stuff like hum, track poorly, odd frequency response, blah blah. I would probably go for an Ortofon or Soundsmith that fits your budget.