Keepers is good. The other thing is to just buy one good one of each variety and propagate that. In my experience currant bushes propagate very easily indeed - as easily as willow.
A couple of years ago OH "accidentally" uprooted a big old unkempt black currant bush behind the old polytunnel. It had already been lying there for almost a week by the time I came home. Naively, I just cut off a load of twigs/branches and stuck them in pots. They all rooted and I now have a row of about twenty black currant bushes which are doing really well.
Granted - the propagation route takes time - but it's very rewarding and you know that you'll have a variety that does well on your soil.
Black and red currant bushes, gooseberries and brambles can readily be propagated by rooting the tips. Pin a few tips down into the soil or large pots and by next year you will have lots of sturdy rooted plants waiting to have their connection to the parent plant cut. I think they will start fruiting sooner than cuttings.
Rasps will pop up all over the place on their own and soon become a nightmare to control.
Rhubarb can be divided (now) although it will take a few years for the roots to be big enough to do that. I know it's not a fruit but it usually has a place in a fruit garden.
It's worth paying for good disease free stock to give you a good start, then bulk up as Suziequeue suggests. I got my originals from Deacons (I think that's the name) on the Isle of Wight - very helpful.
Strawberries are definitely not worth skimping on. Don't accept spare runners from friends and neighbours as they are likely to be carrying disease, Botrytis at the very least, so buy in fresh stock every third year. Once you get strawberry diseases in your soil it's difficult to clean it up. Any of the plant/seed companies will have a good selection which don't cost much.