I have owned and operated a painting and power-washing business in the past. And I purchased, disassembled, and rebuilt a playhouse in the last year.
I power-washed everything. If you are gouging/splintering the wood you are either too close to the wood, using the wrong spray-tip, or that board is rotten and needs to be replaced. I didn't destroy any wood. There were 4 pieces that needed to be replaced but that was because they were rotten or broken. The play-set had high-quality woods, but was neglected and close to 10 years old when I purchased it.
Let everything dry a few days after power-washing. I used solid-color stain, and the more expensive variety. Cheap ones are for siding or fences only. They will not weather well on horizontal surfaces because of water resting there and being a tougher application. Get a deck stain, or log-cabin stain, or whatever seems good to you.
Apply one coat over everything. Apply a second coat on horizontal surfaces where water settles or partly settles, like tops of hand rails, roofs, beams, and decks/walking surfaces. This will give you the best bang for your buck and save a lot of time not doing two coats on surfaces that won't wear quickly anyway. For me solid-color stain was necessary because I had a mixture of old and new wood that would show differently though semi-transparent stain. Plus solid-color stain lasts longer.
Plan on semi-transparent stain lasting 2 to 3 years before needing re-coat (especially on horizontal surfaces). You can get double the life out of solid-color stains. Solid color stains look more like paint, but soak in to the wood and give you some feel for the wood-grain. Plus they don't require priming like paints.