Sounds like you have a great project ahead of you and our climate is perfect for raspberries and blueberries. If you prepare your soil now to plant in fall the plants have a more likely chance of establishing in the rainy season rather than this short period before summer drought.

Raspberries need good drainage & deep planting in our climate kills them, so a mounded bed is a great idea. They need to be in their own separate bed because they spread a lot. Anyone with raspberry growing experience will offer tips to deal with the ever appearing shoots. They shouldn't be close to the blueberries for this reason.

Blueberries need acidic, well drained soil and dislike cultivation around them, so have a plan for weed control. Everbearing strawberries would work as a living mulch around the blueberries. Potted strawberries are available in stores now and bareroot in the spring.

All three love mulch! Wood chips make a nice pathway material between your beds to keep the weeds down and your shoes clean. I often kneel when picking strawberries, so prefer a wider path so my feet have some place to go without crouching sideways.

Is this area full sun? Are there any trees nearby?

If you mow the grass short before it would work. Cardboard is free from stores or restaurants. Overlapping layers work well for smothering grass and help kill turf underneath.

Any place you dig a hole through the cardboard or burlap into the turf before the roots are dead will bring it up to the surface and back to life. If you layer up higher with compost materials or you wait until fall to plant the berries you'd avoid the grass coming back. In my experience it takes a few months before the burlap is decomposed enough for digging a hole through. I prefer the cardboard to smother and the burlap for topping. This method is called interbay mulch and more information is on the soil forum here.

Are there any perennial weeds there like dandelion or clover? Those are more persistent even with smothering, but come out easy enough after this sheet mulching process if they manage to make it to the top. A screwdriver helps you get the root out as long as the ground is wet.

ideas:
+ plan for summer watering of both rasp & blueberries. A soaker hose rather than an overhead sprinkler along the row works well when combined with mulch.
+ more compost - I usually add 3x what I want the height to be as it will shrink down.
+ if possible add manure now that will be composted by fall planting time then no watering will be necessary.
+ if manures are composted (old and cold not smelly OR alpaca or rabbit manures that aren't hot) you can plant an annual vegetable crop in them for summer. Squashes love compost and will spread out over your area. If you don't want squash try planting a cover crop like bush beans in wide rows that will enrich the soil and give you green beans, too. If you don't want to eat the beans just turn them under.

You can always do one part of this fruit area at a time, so you can build the sheet mulch as you gather materials over this growing season. Then add your plants in the fall and more next spring.

Prepare your soil well & your fruits will thrive. Good gardening & fruit eating to you.
Corrine

Here is a link that might be useful: WSU publication on small fruits

Grass will grow through burlap. Use something to kill the grass before planting, then plant in clean soil without having weeds making it more difficult. Spray suitable herbicide or use a smothering layer like old rugs or several thicknesses of damp paper. If you use paper you can mulch over it right after, pull the mulch back to plant after the paper and grass are gone. Do not waste compost on this, put the mulch directly onto the paper or soil. And use a coarser mulch, like arborist wood chips, instead of the dusty, compacting bark most people buy.