It sounds like it is an "everbearing" raspberry, which means it produces fruit on first year canes (primocanes) as well as second year canes. So, the trick is to keep track of which canes have fruited for the second year. Once a cane has fruited for the second year, it will never fruit again, so you cut it to the ground.
Alternatively, you can cut the whole plant to the ground every year (or just mow it) and you will still get a crop of berries from the new canes that year. However, in my experience, the second year canes produce the most fruit, so I'd rather keep them around.
If your winters are very cold, you can bend the canes to the ground and put mulch over them to keep them alive. 10 degrees is about as cold as it ever gets here, so I don't bother mulching and the canes survive just fine.
To produce more plants, just bend some of the taller canes to the ground in the spring and anchor the tip there with a rock or a brick. In a week or two, the tip should root. You can then cut it off and replant it.