Care and Feeding of Raspberries - Help!

Amateur gardener that I am, my general way of doing things has been to swing by the local nursery, look for things that I like, buy them and put them in the ground or a container, then wait for something to happen. Over the last three years - with good guidance from this forum and others - I've become pretty good at getting high quality peppers and tomatoes and learned enough about herbs to keep them alive and to keep my yard from being overrun by mint or catnip.

So today, we move on to the topic of raspberries. Two years ago in early March I bought two raspberry plants and (in normal NeedleCast fashion) stuck them in the ground, dumped good soil around, said "Okay, now make raspberries!" and went back inside until early May. The first year, the bushes developed a bit. One of them made a few handfuls of raspberries and that was about it. Winter came to Baltimore and they died back. I just kinda left them there (didn't cut them back or do anything else). This spring, I added some compost soil around them, fertilized a bit and said "how about enough raspberries for a bowl of cereal this year?"

So, much to my delight, they've done that. Things I've noticed: The "old" growth - the parts of the bushes that grew last year are putting out berries this year. There is also a LOT of new growth that is not making raspberries. From the very limited reading I've done on this, that's normal?

So what SHOULD I be doing that I'm not? Do I cut them back in winter? Do they need trellising/support like tomatoes? Should I just leave them alone?