If any of the weeds have seeds or seed heads on them, do not till them under. They need to be pulled by hand and not put into the compost pile, either. Some weed seeds can stay alive in the soil for up to 10 yrs., so just turning the weed under or spraying the weeds first, will not necessarily eliminate future weeds.
After you've taken care of the weeds first, amend the soil with compost. I like mushroom compost, also, but also like compost that has small bark chips in it. I'll often mix compost half & half with a bag of small or shredded bark. Fine grained compost alone, without bark, seems to turn hard in the ground more quickly. Spread the compost/bark mix about 12" deep on the ground, then plow or hand turn it into the ground to mix with the existing soil.
Raspberries & strawberries are not planted very deeply - plant so that the crowns are just below the soil line. Once raspberries have produced berries, that particular plant (cane) will never bear raspberries again. But that plant will produce additional plants from it's roots. Prune the old cane to the ground (or remove) because it won't ever produce again and just takes up garden space. Raspberries can easily become very invasive, spreading by root under boards, fences, into lawns, etc. Deeper garden edging (min. 6") is best to keep raspberries in bounds.
Strawberries produce new, productive plants by on the ground runners. At the end of each season the original (old) plants should be removed, also, making room for new productive plants.
Since new raspberry plants are produced by below ground roots, and new strawberry plants are produced by above ground runners, I would not use newspaper or weed barriers as it will block the plants ability to reproduce. Newspaper & weed barriers (landscape cloth) work well for perennials but not for berries that require horizontal reproduction in order to produce new, fruit bearing plants.
Blueberries are a shrub so newspaper mulch around them works fine. You should plant 2 different varieties for better pollination, and fertilize with an acid type fertilizer if you live in an area where the soil is not naturally acid, such as Minnesota.
If you use grass clippings for mulch, be sure it hasn't been sprayed with weed killers or pesticides. Don't use grass with seed heads on it, otherwise grass will easily sprout in your berry patches and be hard to remove once it starts spreading. The same applies for hay or straw seed heads. Grass clippings without seed heads works well (if not too deep) for mulch for raspberries, as they can spout up through the grass to reproduce new plants. New strawberry runners cannot develop roots and attach themselves into the soil, if mulched with grass clippings. Grass clippings work well for perennials, though.