Q. Should I stake it or trellis it?
Personally, we (in my childhood family garden) never did and I've never seen or heard of people doing so. That's not to say that some people don't stake or trellis them (maybe in an attempt to try and control their rambling nature). Raspberries definitely do not need to be staked or trellised, just think how they grow naturally in the wild...
Q. How should I prune it?
Depends on the type you have, there are 2 basic types:
- "Summer bearers", these produce one crop a season, on the previous year's canes. New canes appear in the Spring, grow thought the Summer, do nothing during the Winter, then produce their crop the following Summer. Once those canes have produced they can be removed ie Cut down to the ground.
The following year's crop will come from the new canes formed during the previous year.
- "Autumn (Fall) bearers", these produce two crops a year. Green shoots appear throughout the patch in Spring, these grow into long canes, which then bear berries at the tips of those canes during the end of that first season (generally in early Autumn "Fall").
Over the Winter, those canes will turn brown and look dead, but the following Spring, green shoots will appear and produce berries early in the Summer. When that crop run is over, the canes die for real and can be pruned out.
Q. How should it be fertilized?
Depending on how fertile your soil is, I would put down a 2 to 3inch (50 to 75mm) layer of compost, either once a year or twice:
Once a year, put the compost layer down in the Autumn (Fall).
Twice a year, put one compost layer down in the Spring, then the second layer in the Autumn (Fall).
Side note, it's important that raspberries have good drainage.