Here are a few things off the top of my head. Hopefully others can fill in the rest:
1) Get the license plate number, and call your local Police Department's NON-EMERGENCY phone number. Call them and say this: "Hello, this is NOT AN EMERGENCY. I am buying a vehicle and I would like you to run the plate to make sure it has not been reported stolen please." They will be happy you checked with them. If you buy a stolen car, you're up a creek, so check first. Best to do this in advance or in private, so you don't rub the seller the wrong way.
2) Yes, the State of Washington will charge tax. I believe you pay this at the time of registration/title transfer.
3) If you are paying cash, meet in the lobby of a bank. Preferably your bank, but the seller's bank is better than nothing. Ask the seller if they are willing to take a cashier's check. It's up to them.
4) Ask for the VIN and run a carfax check on it. Avoid buying vehicles that have any accidents on the record, and avoid older vehicles that have spent any significant portion of their life in a snowy city that salts the roads. Salt kills cars. I would never buy a car from New England, Minnesota, Denver or Canada. A car from Seattle or a similar city that rarely salts roads is better.
5) The willingness of a private party seller to hold the car while you gather your liquid assets will vary. The general expectation is that "it's not a deal, until it's a deal." If you agree to buy the car for X dollars in five days, be prepared for the fact that if the seller gets an offer equal to or greater than X tomorrow, they will probably sell the car to the person who shows up first with the cash.
6) Haggling etiquette. It is almost expected that a little bit of haggling will go on, but don't be a dick about it. If the car is as-advertised, you should pay within a few hundred of what the ad said. Let's say the ad says the car is in good condition and is for sale for $11,500. You show up and see that it is in good condition, but there's a coffee stain on the back seat. You can offer $11,250 or maybe even $11,100. Don't show up and offer $9000. That's a dick move. On the flip side, if the ad says the car is in mint condition, and you show up and see it's got shitty tires and peeling paint, go ahead and offer something really low. But the bottom line is that if the car is generally in the advertised condition, don't try and undercut the asked for price by more than 5-10%.
7) Run it through Kelly Blue Book, but don't rely on Kelly Blue Book alone. Look at craigslist ads and see what comparable vehicles are actually going for in the area. KBB says they incorporate these regional differences, but they don't compensate enough. Example: Subarus command a premium in Seattle, pickup trucks sell for less here than they do elsewhere. Simple supply and demand.
8) Google known issues on the model car you buy. Example: I like Subaru Outbacks. They need head gaskets and timing chains replaced around 100,000 miles. A car that has already had this done is worth more to me than a car that hasn't had it done yet. Find out if there is a known issue on the car you are going to buy and learn if it's been dealt with yet.
9) Ask the seller why they are selling it. I usually say "If you don't mind my asking, why are you selling it." You want to hear things like "I got a promotion at work, and upgraded to a newer one," or "we're having another baby and need a bigger car." You do not want to hear a bunch of hemming and hawing and subject changing. If it's actually a good car, they probably have a good reason for letting it go. Ask.
10) If anything rubs you the wrong way, walk away. You're in charge. You aren't obligated to anything until you shake hands and exchange money. Trust your gut. You're the boss.
Hope this helps.