No best way, depends what you need, want, and have. Spur or cane pruned grapes. Trailing or erect blackberries. I use a 2 wire system at 3 feet and 5 feet. Posts can be installed numerous ways. I like 4x4 with grapes, and 3/4 inch conduit steel for blackberries. 2-4 wire systems can be used. I also like dual wire systems where I can use them, mostly for raspberries.

Like Drew51 says, there is a lot of ways to build a trellis. Grapes tend to need a strong trellis to support the weight of their fruit which can sometimes be 30-40lbs per vine. Blackberries, depending on the variety, can sometimes get away with a single wire if you tie each cane to the wire individually.

What kind of blackberries (e.g. variety or varieties) did you purchase?

If I were you for the Blackberries if you are planting them in a row, I would sink an 8 feet long 4X4 into the ground 2 feet, anchored in cement on each end. Run a 12 gage wire at the 2 foot, 3.5 foot, and 5 foot levels and pull tight. Use 6 foot long metal posts between each plant sunk 1 foot into the ground and tied to each wire. For the grapes I would use a "T" trellis made from 8-foot 4X4 posts and 2X6 cross pieces with 3 12 gage wires connecting the cross pieces, pulled tight. You might need anchors for your end posts to prevent leaning even if you use concrete in the holes, which I did.

CharlieBoring,

The blackberry trellis you build should reflect the type of blackberry you are growing. If you are growing an erect variety like Prime Ark45 or Oauchita, you really only need a single wire at 5 ft to keep the plants from blowing down in a wind storm. Otherwise, they would be fine without the wire. If you were growing a trailing or semi-trailing variety like Triple crown, then you really need a heavier trellis to support the weight of the plant plus fruit and give the trailing canes something to keep them off the ground.

What kind of blackberries? What kind of grapes? What zone are you in? You might take a look at every one of the people who responded and see that next to their name is the agricultural zone they're in. Since you're posting on a gardening site, why don't you show the zone or better still, the zone and what part of the country if it makes a difference?