Pickaroon is great, and it's not that far - at most an hour and a half off the pavement from Foxpark. The road is rough and narrow with switchbacks going down to the river. You'll want to take your time. When you get to the bottom of the hill at the river, the road will turn left, and you can follow the road up the river for about a mile and a half to the Douglas Creek confluence. Scattered camping is all along the river, but the best sites are right below the confluence - lots of grass and cottonwoods, tables, tent sites, parking, and a pit toilet.
You can also follow the road up Douglas Creek for another mile or so, with scattered campsites along the road, to the Douglas Creek trailhead into the Platte River Wilderness area. If there's still enough water in Douglas, it's great fishing - if not, fish right below the confluence.
Use tapers off significantly after Labor Day, but it still could be quite busy on a weekend - prime spots might fill up early on Fridays. You do not want to come in from the west side - you can make it with a 4wd, but it's mainly used as a launch for the commercial guides - camping there would not be good.