We've done the trip multiple times in both directions, and like both -- but they offer vastly different experiences!
Travelling the East side of the peninsula, you're going to travel through what I think of as "cottage country" -- lots of small communities with long stretches of mostly-built-up rural residential areas, particularly along the shoreline stretches of Puget Sound western shore. It feels like lots of summer places for city residents? It's picturesque but quite populated.
Camping is sparse, in my memory.
And once you get to the Southern end of Puget Sound, say around Olympia, you're in big city & freeway country.
Depending on where you're headed, from there you can stick to major highways & freeways generally South or SouthEast, or head Westwards through Raymond & Aberdeen, which is mostly forest-based industrial, and agricultural, country. Think sawmills & farms.
The Western side, once you get past Sequim & Port Angeles, is forestry extraction country, with long stretches of interior highway through big timber, interspersed (eventually) with short but beautifully scenic stretches along the actual coastline. At the South end, eventually also ends up near the Raymond/Aberdeen area.
Along this route, I particularly like the drive along Crescent Lake, and camping at Kalaloch campground (if you can get in!). Somewhat lengthy but worthwhile side trips, depending on your time, are out to Neah Bay and to La Push/Rialto Beach, both on the ocean.
Great campsite at Dungeness Spit in Sequim. And if you go that direction, one of my favourite breakfasts ever is the Dutch Apple Pancake at the Oak Table in Sequim.
It's all great, but two very different experiences on the two sides. Have a great trip.
Lotar & Wendy
"Sit loosely in the saddle of life" (Robert Louis Stevenson)