Can I plant them all together or will they cross pollinate?
How do I plant them?
How much water do they need?
Will they be ok in my clay soil or is my only option to let them die?

This list of questions could go on for awhile.... any information will be extremely helpful. For now.. I'll leave them in their pots and water them... the extra water will just run out of the bottom, right?

Depends on the variety somewhat, but in my experience red raspberries are upright canes from 4' to 6' tall and they can sucker like crazy. Black raspberries (at least the Wyoming Black I grew) formed long arching canes up to 8' or 9' and did not sucker. Blackberries can be upright canes or huge arching canes up to 15' or more, depending on the variety. My "Chester" blackberries produce almost no suckers, but I'm in kind of a cold zone for them.

Clay isn't ideal, but they are tough and will survive and produce berries. I don't water my raspberries. Sometimes I water the blackberries, but only if it's been very dry.

The raspberries too but will do better with a more compost mixed in with the soil and more water.

You only need to water if you get less than 1 rain week and immediately after you plant.

Keep weeds away from them my mulching with oak leaves or other tree leaves if you can get them. Otherwise a thin layer of pine needles or straw will do.

If you keep the weeds away you'll get new plants every year. It's required to get new raspberry plants every year if you want to continue to get raspberries as they only bear one new wood once and that cane stops bearing and usually dies over the winter. The raspberry then come from new runners.

You can use fertilizers in getting them established but by next year if you mulch with leaves you definitely won't need to.

I live in 6B and both blackberries and Cumberland raspberries ('black raspberries') grow wild here. The blackberries typically in hot, dry sunny locations and the black raspberries in cooler, shadier locations right at the edges of forests with richer dirt than the blackberries and with leaves serving as mulch and fertilizers.

I don't grow black raspberries, but blackberries can grow up to 15-20 feet, along a fenceline is good because you can train them along the fence in a wide sweeping arch. If you allow the tip to hit the ground it will root and make new plants there.

raspberries can grow just in the plain dirt without supports, but I find mine need them when fruiting because they tend to flop over. The new green growth (next year's fruit) doesn't need supports.

You probably won't get any fruit this year but you will next year! Give each plant lots of room in between because raspberries love to fill in. I let mine overgrow (I try to keep it to 5-6 plants per sq ft but I failed this year to cut out enough) and its a jungle in there.

I planted Triple Crown blackberries so they are trailing, do need support (planted in 2012 and they didn't need much/didn't produce much in 2013 but need trellis now). Those I intentionally tip-rooted to get them to spread, cut the canes connecting old and new plants this spring once the new plants started sending up shoots of their own (not just roots on the buried tip). The older canes are also sending out fruiting laterals now.

The Encore and Killarney raspberries are more upright. Those varieties you want to prune out the canes after they have borne fruit since they only bear once. The new shoots that are coming up now will bear next summer. But you have to check to see if you have a different type that will bear twice on the same cane before deciding how to prune them.

I did plant them where they get some late afternoon shade but found a lot of bleached-out raspberries last year - hopefully this year will be better, we had a heat wave last July and while I kept them watered, the sun was intense.

We covered the bare soil with burlap on the sides and between canes, some clover and weeds have managed to seed in the gaps but they're easily pulled, I do check each spring to see if new canes or weeds are coming up under the burlap. If canes, I rearrange the burlap to give them light and let them get stronger before transplanting (try to keep them attached to mother plant roots and just move the new shoot a little into the row, rather than cut), if weeds they are weak and I can pull them. We've mulched with wood chips from tree service in between rows and that hasn't worked as well - the potentilla is spreading in the mulch anyway.

Here is one way to avoid that. tear off a strip of paper towel and stick it up into a hole, then sit the pot on a whole paper towel so that the strip in the bottom of the pot sits on the whole one, (most of the whole one is open to the air) this will help wick away the excess water and have it evaporate away.

To get your mind around this think of a sponge. It hold a lot of water just hanging there in the air. but set it on a paper towel and the water wicks out, leaving the sponge very damp, but not heavily soaked up.

If it were me, I would take the red raspberries and plant them in my back yard, the black raspberries and black berries I would pass on to someone else. But, that is not what you want to do.. The reds will produce a crop on every stalk every year and be pretty much trouble free. The other grow the stalk on year one and grow berries on year two. Birds love black berries, but leave red raspberries alone.

My black berries are on probation this year. They are at risk of being dug out and replaced with a pear tree. Per Olpea's advice I am not letting the black berries get above 4' this year as a way of getting better control over the netting and etc so that I can get all the berries and at full ripe.

That is not true at all, it so far off, it's actually funny! I can tell you obviously have never grown them.
Raspberries can bear fruit for 2 summers, unless summer bearing. Which are not that popular. Most raspberries sold are everbearing and can bear for 2 years. The crowns do not die, they just produce more canes, and runners too.
Which can become new crowns.
Google "how to grow raspberries" to understand the life cycle, and what is the best way to grow them. You can contact your local extension office which can give you very good info for your area.