I think you need to start with an understanding of how raspberriess grow. They don't form a permanent shrub. Those smaller canes growing around the large ones will give you next years fruit. The large ones get cut down. Google primocane and floricane. Raspberries are not like regular shrubs and they are pruned differently. They grow new canes from the ground every year. To know how to deal with them you need to know what type you have. If you have summer fruiting raspberries (floricanes) you cut out all the canes which have fruited and leave all the new unfruited canes behind. They'll fruit next year. If you have autumn fruiting types (primocanes) you cut all the canes to the ground in the early spring.

Storing canes in the basement under lights is a waste of your time and money and will be deleterious to the canes. Just dig them up and move them when you want to. But water well. If you can't replant them immediately just pot them up and keep the pots in the yard somewhere. What you read about storing canes somewhere refers to dormant, leafless canes, not actively growing ones.

We have Summer bearing red raspberries. I've just let the canes sprout wherever they wanted and cut down the old ones after they bore fruit. But this year the bush has gotten out of control with new canes invading the middle part of the backyard. Some are up on a small incline we have in the back of our yard and I think they are ok to stay there. But this spot has had Raspberries growing there for probably 5 or more years and I think that they may have sucked up a lot of the nutrients and need a new spot to grow. I'm just unsure of how to organize them. I've dug up and moved runners before but now there are so many I think I'm a bit overwhelmed and not sure where to start tackling them.

So dig out as many unfruited canes as you want to keep and either transplant directly to where you want them or pot them up and put them in an out of the way spot until you can plant them. Definitely don't put them in the basement. To keep them in bounds you could just mow around the patch when you do the lawn. If you are starting a new planting you might find it easier to keep them in bounds if you plant in a row with a post and wire support and keep up the maintenance pruning and training. Commercial growers do replace their canes every few years or so but in a domestic situation I think you can leave them longer. Are they showing signs of deterioration? If you haven't been restricting the number of canes that could be a problem.

Thanks, clearing up I see I have a lot of small canes I will dig up and move to a new spot in the backyard. I haven't restricted them really so I want to start in a new spot where I can restrict them and stay on top cuz those buggers like to spread out!