Jump summed it up pretty well, IMO.
Next week in the area, the temps are supposed to be on warming trend, but still cool...Spokane is in the hottest place in the valley. I only go through there on my way to GEG.
Unless there is some kind of event there you want to attend, such as an orchestra, or other city event, like the other sentiments here, I would advise getting on I-90 and put the pedal to due East.
The bigger the mountains, the better, so my thoughts always include those.
As JumpZ said, and to me too, the worthy places take some time and driving to get there , so be prepared to drive, in that case. The panhandle of Idaho from Spokane is around 60 miles wide, if I recall correctly, then once you hit Montana border, it isn't called big sky country for nothing.
One place I feel is worth the drive to see are the giant cedars ...one spot is in the Idaho Avery/Pritchard area . If you go that far, a stop at a very good place to eat, the Snakepit, is a must. Don't worry about the appetizing name.
The hills around that area are all Natl. Forest , and full of history. Lots of pull outs off the main hiway that have historical signs to read while enjoying the views.
The other place is Ross Creek Cedars , in NW Montana . That is probably too much of a drive , but either place is worth seeing those massive one of a kind trees.
I would say either contact the info centers in Idaho, East WA., and see what things you can pull up, according to your interest and time. Take note of driving distances , and any weather changes . Next week is supposed to be fairly stable, with no big storms expected.
Have fun, but as in any city , be self aware of everything around you.