"Suckers", usually a negative term in gardening, are desirable in caneberries as that is where future year's fruit comes from; better to call them protocanes.

If the new canes are numerous, you can remove up to as many as will leave you with a similar number of old and new canes. The thinning will not cause more canes to form this year.

If you have sufficient new canes sprouting, anything coming from runners can be dug out, do not dig too close to the main clumps, stay a foot or more away.

Sanitation is another topic that is better re-posted in a separate forum question.

You should learn what type of Raspberry you planted Primocane or Floricane. Primocane produce a small crop on first year growth and a larger crop on second year grow. After the second year the old canes are removed and leaving the first year cane. Removing your suckers will leave you with no fruit and a dead plant. Floricane berries produce only on second year cane. Once again they should be removed after fruiting. Its not uncommon to get a little fruit first year on Floricane so don't just assume.

They're all primocane, I specifically ordered everbearing varieties Anne and Carolina.

Thanks for the advice!

Good choices! I believe it is just Caroline. Harry Swartz tended to name varieties after the women in his life.

To get a florocane crop thin previous year's primocane brambles to about 8-10" spacing between them. Some even space more to concentrate more on the fall crop. That might be a better choice for you since Caroline is an early fall primocane producer. I would not thin primocane shoots(personal choice) because you can create more cane injury in the process. Just mow edges of the row to keep bed manageable.