Welcome, Apewheel!
The dried out cane is probably the original piece that was used to start the new plant. If it is fully dry, I would cut it off, but if it still has some juice, I would keep it, as the developing plant will use it as a "food bank" in tough times. A raspberry plant needs a large container, if you are going that route--about 2 ft. wide by 2 ft. tall--so the wine barrel would work well for the single plant, for as long as it lasts. If you are gardening on the coast, you could keep the container in full sun. Hotter areas, such as Riverside, eastern San Diego County, or the desert, mean keeping it where it will get afternoon shade, in the summer at least. The plant will produce more and larger of those "vines" from below ground near the base until it forms a fair sized clump. One plant can eventually grow into an entire patch! Whether you do one clump, or a row or patch depends on how many raspberries you want to grow. 'Willamette' is a summer-bearing type of raspberry, which means that it flowers in spring on wood that grew the previous year, then the flowers develop into the berries that ripen in summer--early summer if your climate warms up soon in spring, or late summer on the cooler parts of the coast. That means that the cane you have now will produce a few flowers, this spring, and the canes that grow in 2011 will bloom and bear in 2012. If you can stand to do it, it would be better to pinch off the flowers on the first cane, to put more energy into good roots and canes for 2012's crop. Since you can grow plumerias outside, I would plant it as soon as possible, to let it grow roots before summer's heat starts. Be sure to water deeply and regularly, especially through the summer, and even more so in a container. Feeding should be monthly from February through November in a container, or three times a year--Valentine's Day, Memorial Day, and Labor Day--in the ground. With alkaline soil or water (most common in the west) a 3-1-2 proportioned complete fertilizer with extra sulfur and micro nutrients is ideal. In neutral conditions (occasional on the coast) a 2-1-1 formula is more appropriate, since phosphate dissolves more quickly there. Raspberries are semi self supporting. but tying them up will help keep them from lashing around in the wind and ripping up themselves, other plants, and the gardener!
I hope all this helps!

Tug, wow thank you so much for all the information! I've been so busy it's still in it's container :o but I'll try to plant it over the weekend. I will print up all your advice and I won't mind pinching off the flowers this time around for better results the following year. I really appreciate your help. I'm looking forward to learning more on this site. I have lots of plants and they all do well, but I'm only just getting into growing actual food. I don't know what I'm doing yet but I guess I'll learn as I go along. It's nice to know there's a helpful community here to help me along. Thanks again.
-a

Don't stress, A! Most problems are fixable. If worst comes to worst, well, I've been gardening for forty-odd years, and I've killed at least one of everything that I know how to grow. Good luck!
Tug