Hands down I would say Wade's Indoor Shooting range in Bellevue, Washington. Just East of Seattle about 20 minutes.
http://www.wadesguns.com/
Why? There are lots of indoor and outdoor ranges in Washington State but why my favorite - for the staff, the range rules, attached gunstore and gunsmith and also for fun shooting.
The Staff - very helpful, friendly and willing to teach and make sure your safe.
Range Rules - every range has it's particular rules from no cell phones to no picking your nose. It runs the gammit and they're all different besides your safety rules. At Wade's you can shoot as fast as you can so long as you can keep it on paper. Not many ranges will let you do that. On a side note: You can also shoot their full auto's too (for rent of course). Wade's does have a particular rule - you can't rent a gun if you are alone (stems from people walking in and renting a gun and going to the bathroom to do themselves in). *There are different variants out there I'm sure. From not being able to rent unless you bring in a gun at another range I've been too.
Attached gunstore - great for looking at what's new and dreaming a bit on a shooting break. Also has loads of gear and ammo to look at too.
Fun - It's not just a shooting range - it has good lighting and good air flow to clear gunpowder in the air. They also have classes ranging from new shooter education to advanced tactics.
Yes, there are many ranges in Washington both private and public but overall, Wade's has my vote for an overall positive experience and I'm fortunate to live close by (30 minutes).
Disclaimer: No, I do not work for Wade's :-)