A throw rope is the first thing you want to carry. There are several brands and styles, and after time you will learn what you like and what meets your needs the best. Buy a quality 50' bag, and practice often with it. Ropes are much like guns, in the correct hands they will save a life, in the wrong hands they will take a life.
NRS has a series of rescue videos on their website, I would recommend watching them. Practice first throwing the rope when it is properly stuffed, and then learn how to quickly coil and throw a rope for your second shot.
Why did I say a 50 foot bag? In the classes I have attended, most people can throw a bag about 40 feet, 50 feet with training and practice. Not many people can fully deploy a 75 foot bag. They do have their place, and you will learn that over time.
Throw ropes are primarily used to rescue another person, and people are always the priority. Some throw ropes are designed to be used as haul line in a Z drag setup to unpin a raft or kayak, but not all are.
As you have read on here, try not to lose a rope. One of my old paddling buddies lost a friend to a discarded actor rope on a class II section of river that I frequent to this day.
If you can find one, join a club. I just joined a club 2 seasons ago, and even though I have been going since I was a kid, I am learning more and more at a faster rate. It is a group of well trained individuals that allow me to push my limits more safely, while I get to contribute to their safety as well. It also allows me to have a larger network of friends to paddle with. I have a long list of people I can call on now to plan river trips.
Wishing I was on the river instead of surfing the web...