On foot: Within town no place is much more than a half-hour walk away, an hour at the most, and many things to do are easily found in the small downtown area. The streets are pedestrian and bicycle friendly - with sidewalks and bike lanes on most major roads, and low traffic on others.
Best Biking Around: Great biking through surrounding vineyards and wheatfields. To Waitsburg and Dayton, to Milton Freewater and Touchet, numerous 15-60+ mile routes. But mind those hills! Stop into the following bike shops for info on routes and rides:
Bicycle Barn
Allegro Cyclery
The Valley Transit (http://www.valleytransit.com/) local bus service runs in the daytime around town, costs only 75 cents a trip with free transfers, and also provides bike racks on most buses. It's main transfer station is located downtown, and it also runs to the airport.
With many peaceful parks and quiet streets lined with large old, shade trees to wander along and admire the charming neighborhoods, leaving the car behind is more than plausible.
With wheels: Bring your own car to drive around town or out on the country roads with, or rent one from one of four available companies easily found online. Even if you plan to explore mostly on foot, having a car can be a boon for a trip up Mill Creek or getting to some of the wineries located around the airport and out in the country.