What's the best neighborhood in Spokane for a first time visitor?

I'm on here trying to figure out my trip to SF tomorrow....so funny. Saw your post and here is my advice as I have been here my whole life (thankfully I travel to keep me happy here.) I would stay downtown, there is so much more to do and see.  Lots of fun areas to walk around to.  Kendall Yards is nearby and is up and coming...not a lot there but great dining spots.  Good place to rent a bike and be sure to hit Brain Freeze Creamery for ice cream. Staying downtown puts you by Riverfront Park which is fun to go through...many bridges over the river with great views.  Take a ride on the gondola to get a close up of the falls.
Any of the Davenport properties are a home run. (Really nice one just opened up, it is gorgeous.)  No matter what plan to at least walk through the old historic Davenport.  Then the sister property nearby is a great place to get a drink/bite to eat for happy hour.  They can run a bit high so to watch $ you could stay with any chain hotel downtown.  
Spokane is pretty easy to figure out compared to LA.  Basically when you get in at the airport, you are in Airway Heights, just West of Spokane, 10 minute drive puts you downtown.  
*Stay on the freeway heading east about 20 minutes and you will be in the Spokane Valley (boring).  But! if you continue driving another 35-40 minutes from the valley you'll be in on to Coeur d'Alene, Idaho which is so fun this time of year.  Public beach and you can rent paddle boards,etc. Great little resort town! Hike Tubbs Hill to find your own way down to the lake if you want a fun easy hike.  
* Take a downtown Spokane exit south and you'll be in an area called the South Hill. There is a fun area called the South Perry District is there, worth a trip.  Lots of cool historic houses and old money make up the south hill.
*Take a downtown Spokane exit and go north, you will just be on your way to normal stores/shopping. 
Also downtown, I love Zola.  Look them up online and see what music is playing, changes nightly.  Low key and so fun.  Next door is Boots Bakery & Lounge which is open late with an awesome vegan menu of goodie.  Across the street is the Saranac Public House...during the day you will find a few antique stores and my favorite boutique is close too, called Lolo Boutique.   
Have fun!

The Davenport Hotel or The Davenport Grand, Autograph Collection are 2 nice hotels. Valet parking and security, central location

The Centennial Hotel, Downtown Spokane.

The River Park Square

The Davenport Hotel hands down

My recommended sites/stays would be: Visit & stay Downtown Spokane - in general. For specific site seeing, you have to stop at Auntie's Bookstore at some point, and take a walk through Riverfront Park. Stay at the downtown hotel The Davenport Hotel. (If affordable. There are cheaper options.) And/or in the neighborhood right off of the downtown area, Browne's Addition. 
Granted, it's been a hot minute since I've been there. But I trust that these will still have the feeling of history, beauty and warmth that they held when I was living there. 
Have a fun trip!

The Davenport Grand, Autograph Collection is near the Spokane Convention Center, Riverfront Park, and the Centennial Trail.

I highly recommend staying at the historic The Davenport Hotel. It's in the center of downtown. Many restaurants, you can walk to the park and places to shop.

Stay at Mirabeau Park Hotel & Convention Center. It gives you central access to great views, food and locations without all the traffic. You can even take a bus if you don't want to drive as it's safe & reliable.