Sell EVERYTHING you own. Ikea furniture is commodity stuff on Craigslist. You can sell your dresser, say, for $20 and buy the exact same model when you get to your destination for about the same price. It's like a free moving service! We did exactly this with our futon, we have the same model as we had in DC and I paid the same amount to buy it here as I sold the old one for.
Anything you can't sell, donate it to Goodwill or Salvation Army. Likewise, anything you want to replace when you get there but can't find on Craigslist, go to Goodwill or Salvation Army.
My wife and I moved across the country last summer and we brought a ton of stuff that was just stupid to bring. Go to Goodwill and Salvation Army in your area and see what kinds of stuff they have for sale, since this will help you understand what you should bring and what you should just get rid of. Some items can be harder to find at thrift stores, but others are plentiful. Small kitchen appliances are cheap. Plates/bowls/glasses are cheap. Dish drying racks, silverware, etc. are all easy and cheap to get.
Now, for anything that's non-negotiable and you absolutely want to bring with you: Amtrak will ship stuff for you for the lowest cost. From Washington DC to Seattle it cost us ~$0.60 per pound for a pallet of boxes. Call or check their website for details. Your pallet will get dusty/dirty on the train and you can't pack fragile items (they warn you about this and it's true, a few of our things broke in shipment). We used Amtrak for a bunch, but had more stuff that we ended up shipping UPS. This got VERY expensive, especially for any oversized items. One box with skis in it I think cost us $70 to ship! My desktop computer was similarly expensive because it's so heavy. I think we paid about $500 to ship 20 boxes. This is why you want to get rid of as much as you possibly can, most of it is cheaper to buy than to ship across the country.