my wife and i are staying at a hostel in august that is close to pike place market. if we want to check out cool places in seattle for 4 days or so, should we rent a car when we fly in, or can we rely on public transportation without problem?

Public transportation imo. Buses are numerous and run often. Once you get to an area it's best to walk around anyways and traffic and parking are a bit crazy in Seattle. Seems like they have outgrown their roads.

depends on what your going to do if you are staying in seattle.
for your sight seeing the buses are great if you are going outside of seattle proper it can take a while to get there by bus and it usually isn't direct.
but in full disclosure i don't drive and use the buses to get where i am going.

Buses for sure. Great public transportation - you should be able to get anywhere with one transfer at the most. And you've got a lot within walking distance - all of downtown and Belltown. You can walk to Capital Hill/Broadway or down to the ID - it's a bit of a walk, but not bad. Buses to Fremont, Greenwood, Green Lake, Georgetown, U-District. Maybe rent bikes? You can take the bikes on the buses for the longer distances, and then cruise from the bus to spots in between pretty quick.

The price of parking a car downtown is another blow against renting a car. (I doubt the hostel has free overnight parking.) If you decide to go to Rainier or up to the nearby mountains, you can always rent a car for 1 day. If you are just doing the urban scene, stick to the bus & your feet. You can also take the light rail from the airport to downtown for $2.50 ea. (Look up METRO online for where to get off - probably Westlake.) Also, if you want a cheap boat ride out on Puget Sound, take the ferry to Bainbridge Island or Bremerton. Just as good as the tourist boats for 1/3 the cost.

Yeah i used the buses too when I was there 2 or 3 years ago. But get yourself a schedule, and keep in mind that come evening, some of those buses run very infrequently! In general, it's a very walkable city, and the buses can get you from main area to main area. But it would be nice if they ran more frequently in the evening, since that might limit your mobility.

Going further back, I think I stayed at that hostel, but it would have been more than 10 years ago. Back then, it was OK as I remember.

Have a great trip!