I'm not sure I understand the difference between canes from the base, and root suckers? The primocanes can be thinned, but best to wait till they are larger, at least 18 inches. Or just wait till the end of the season. You want canes to be about 6 inches apart or farther, no closer. Raspberry plants this year, i think in response to the cold winter are all throwing a large amount of primocanes. And yes you can trim down the floricanes now.
Just to be clear, removing the suckers, will result in the plant throwing more out if they are small, wait till the are a foot and half, or till end of season. I would keep them if 6 inches apart. It's a raspberry patch afterall.
Once floricanes are done fruiting cut them out completely.
You can pull the rooted tips out too, cut the canes down some, so they don't keep doing it. You have to watch blacks closer because of the root tipping. it sure makes making new plants easy though!

If they're black raspberries they shouldn't be sending up any suckers- could they be coming from red raspberries nearby?

I would get rid of all the tip rootings that you don't want to turn into full plants, otherwise it'll be a jungle soon.

As far as pruning, while cutting back the laterals is often recommended it's not required; the main thing would be to avoid having fruit lying on the ground if the branches are too long.

Thank you. When I went out, clippers in hand, and cut back the floricanes it became clear that all the suckers had rooted from the tips, mostly last year's laterals.. I guess I should have cut back the canes a lot more last fall. Anyway, I cut them back now. These are only 2nd year plants, so I don't want them to set too much fruit yet.

I assume I should dig out all those tip suckers. I think I'll try to save and plant 1-2 elsewhere, but I figure they shouldn't stay where they are.

Here's a picture of the new canes coming up. They're as thick as my finger.

I left the longest laterals about 18". I hope I did right!

Oh--what do I do about feeding?
Liz

2-4 inches of composted manure every year or second year will do fine.

Young caneberries don't care how much fruit they set, you don't have to prune to limit fruit. That is for young blueberry or orchard fruit trees.

Black raspberries also don't care how close the new shoots are; commercial fields have a couple dozen per clump. You may want to thin them if there are more than you care to handle, but a tight grouping of 3 is no problem.

The fruit does tend to get smaller the farther down the lateral you go.