if you buy container grown bushes then they can be planted any time provided the ground isn't frozen or waterlogged. March is often a good time. as for which fruit that is personal choice. But Blueberries Raspberries and red/white/black currants are all thornless. I suspect Logan berries are too. There may well be a gooseberry that's thornless by now [don't know personally]

If the soil is waterlogged dig in grit etc to improve drainage before planting.

There is a thornless Logan berry, also Boysen and Blackberries.

There are a few thornless gooseberries. I haven't grown them but there is info on google about which are good ones. There's also a good description of thornless blackberries, (on another site) which tells you which are more vigorous too.

You have quite a few possibilities. I love them all. The sooner you can get them into the ground, the better. Try to plant before they break dormancy but as soon as the ground is workable - late February/early March.