First year canes grow.
Second year canes develop side shoots with flower buds.
Third year canes may develop side shoots in a lot less quantity (diminished crop), but in domesticated Raspberries usually die.
Removal is the best policy.
Unless you're dealing with wild black cap Rasberries, this is the standard growth cycle.
You can propagate by letting the canes bend over in a U till the tip touches the ground in the second year.
They react by creating a "dragon head" which roots and clones the plant.
The new growth from this follows the same routine, but may allow for new growth off the attached cane to produce berries.
Domesticated Raspberries are prone to various fungal diseases and depending on the cultivar also do not tolerate Anthracnose virus.
Dispose of the trimmings, you don't want either to be serving as mulch for your new canes.
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Trim the old canes close to the ground.
First, you don't want the plant to invest any of its energy into maintaining these old canes.
You want the energy to go into the new canes.
Second, if you leave them, they will dry out and may rot in wet weather.
With decreased air circulation, the rot may spread to the healthy young canes.
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Another thing to keep in mind is older canes may harbor insect pests (borers, for example) and allow them to overwinter in your raspberry patch, making the next year's fight against infestation a lot harder.
So, not only prune the old canes, but REMOVE them far away from the growing site.
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Where you specifically noted your everbearing variety of berry, I think some of the previous answers are missing something important.
While you won't get fruit a second time on the end of previous year's canes, you don't have to cut the canes to the ground after their fruiting season.
There are some situations where it's pretty much a must, but that depends on your local climate and when you want next year's crop harvest.
If you want a July fruiting (note that it will not be as full of fruit as your Fall fruiting)
you can cut back the canes to just below where they fruited in the previous Fall.
The same cane will produce fruit its second year from below where fruit was produced in the first year.
After that fruiting, cut the cane to within 2" of the ground.
This will delay your Fall fruiting by 1-2 weeks, so you have to be careful if you live in an area with early/cool/wet Fall seasons that it doesn't mean that you miss your Fall harvest due to things like fruit rot.
For more info, you can read this page by UConn Extension or this article by Fine Gardening.
The pruning-specific part of the Fine Gardening article is on page 4.
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You've got Primocane raspberries there.
Those produce on this year's canes - as opposed to Floricanes which produce on the canes from last year.
Technically, there is no requirement to prune them back.
You could most definitely allow them to fruit next summer
but I believe the general consensus is that Primocane-fruiting raspberries are chopped back to the ground.
Here in Virginia, I'd cut the Primocanes back in January and I'd cut them back to the ground.
If you choose not to prune those canes, you'll get some fruit on those canes next year
but I don't think that's a common practice and
it, frankly, complicates an otherwise simple raspberry growing experience.
I think Primocanes are a great choice for the home raspberry grower from the simplicity standpoint.
The raspberry canes grow and fruit in the same year
then you wait until it's dormant and cut it back.
Next spring it'll send up new canes the process repeats.
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I just bought a Fall Gold Raspberry plant. It came with instructions to prune away canes that have already fruited because only new canes can fruit. What If I were to leave the old canes there? Do the canes serve any other purpose to the plant holistically other than fruiting? Will leaving the old canes there harm the plant, or even benefit the plant, or not affect the plant at all?
Are old raspberry canes required to be chopped off?
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<pruning><raspberries>
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There is no requirement to prune them back but you don't want the plant to waste energy on these old canes. Technically, there is no requirement to prune them back but removal is the best policy. You can trim the old canes close to the ground. You don't want the plant to invest any of its energy into maintaining these old canes. In addition if you leave them they may rot and may harbour insect pests. On the other hand if you choose  not to prune those canes, you may get some fruit on those canes next year
Technically, there is no requirement to prune them back but removal is the best policy. You can trim the old canes close to the ground. You don't want the plant to invest any of its energy into maintaining these old canes. In addition if you leave them they may rot and may harbour insect pests. On the other hand if you choose  not to prune those canes, you may get some fruit on those canes next year