I'm not sure what can be done online and what has to be done in person, but you can probably get your car registered as well as your voters registration done at your county auditor's office. Unless you are going to King or Thurston county it should be easy, I've never waited more than a few minutes in Clark county.

You will really like how voting is done in Washington compared to most of the rest of the country. All voting is done by mail-in ballot, if you lose your ballot you can go to the county auditor's office and get a new one. The ballot comes with a return envelope that doesn't require a stamp, just drop it in the mail by election day and you're covered however there are ballot drop boxes located around your county. Also around the same time the primary ballots are mailed every address with a registered voter will receive a voters guide, this has every candidate and every ballot measure for the whole county, even ones you won't be able to vote in. Also there is no party registration in Washington, all primaries are open and the top 2 vote getters move onto the general election.

Back to your actual questions. You will probably have to go into the DMV to get a new license, I believe most of them offer appointments so schedule in advance and save time.

If you only want to call your insurance company once you can probably wait until after you get your car registered, however if the price difference is that significant you may want to call before you move and tell them the date you will be in Washington and they can make it effective in advance for you.

I don't believe that the order on any of those 3 items are critical to the state, but the costs involved with insurance may make that one more critical to you. The license and registration are both supposed to be handled in either 15 or 30 days after moving.

Edit: reading some of the other replies it appears you are probably supposed to get your license before registration, but I find it hard to believe that it is required as I know it's possible to get a Washington registration while living in another state but that may be just for a commercial vehicle.