Liberty Lake is nice, because there's a big (thousands of acres big) park at the south end. Also a popular swimming hole in the summer. There's a six mile or so loop trail in there. In the winter, you can go snowshoeing there and might only see a couple other people the whole time. In the spring, lots of deer are around, and I saw a couple of moose this year too.
To the north, just across the freeway is the river with the Centennial bike trail, which is paved and runs more than 30 miles, most of it along the water.
A possible downside (depending on your interests) with Liberty Lake is it's kind of a sleepy bedroom community without much in the way of nightlife or restaurants (though there are some, and it is definitely growing).
Also in Spokane Valley, closer to Spokane, the Dishman Hills area offers a great park with miles of crisscrossing trails (I actually kind of prefer this one for dog walking, because there are a lot of different choices, and you can swirl around different ways each time). Here, you're only a few minutes from downtown via the freeway, and again the bike trail is not far to the north.
For another really big park, check out Riverside State Park which borders the northwest side of Spokane. That bike trail is over there, too, and there are hiking trails you can meander along. I'd say this is the most lushly green section of town, though I think it also gets the most snow in the winter.
In all these areas, you can get a pretty nice 1 bedroom condo for 150k (and a not-entirely-hideous one for closer to 100k). Liberty Lake is probably the most expensive of the areas as far as median house prices.
Traffic is pretty light most of the time in Spokane, and I (living on the South Hill) can get to any of the areas I mentioned in about 20 minutes, so wherever you choose, trails aren't far off.
