Seattle Center, Seattle
Why spend your travel dollars and travel time on some goofy ride next to the space needle. Unless you don't really have the opportunity to go on these kind of rides where you are living, just don't do it, wait until you get back home. They are expensive and you are supposed to be seeing and doing things you can't normally do where you live.
Seattle Center sounds like it should be the hub of activity in Seattle. It really isn't. It's just a fairgrounds without the fair, except on special occasions like Bumbershoot or Bite of Seattle, which are actually really fun and recommended. When a festival is not going on, the place is kind of dull, parking can be expensive, it's quite a walk from all the other downtown attractions, and the fun things to do require separate admission that will add up fast. You'll find yourself asking "is this all it it is?" Because it's really just a city park with some tourist-traps like the EMP and a small carnival. And unlike other parks in Seattle, there isn't any scenery or impressive views to enjoy unless you pay to go up the Space Needle for a few minutes.
Unique Suggestions: Try to go during a festival like Bumbershoot or Bite of Seattle. You would have to pay admission, but at least you would get the real Seattle Center experience as it was intended to be enjoyed. (It was built as the festival grounds for the World's Fair in 1962.) During such festivals you can experience music, food and vendors in a fun, family friendly environment. On these occassions, Seattle Center is actually a cultural hub rather than a tourist trap. (See http://www.seattlecenter.com/events/ for the events calendar.)
The Space Needle restaurant is actually really good, and it's not too expensive if you go for lunch. If you eat there, you get to enjoy the view and great food without paying admission like everyone else pays for the observation deck.
The other great thing to do is watch children play in the International Fountain on a hot day. The fountain randomly spurts water, which scares the children, who scream and then run back for more. If you have kids, you might consider bringing their swimsuits.
If you're willing to spend the money, you could go to the Pacific Science Center to catch an Imax film or the EMP if you are into looking at rock star memorabilia in glass cases. Some people really enjoy that sort of thing, especially if you have kids that would enjoy the interactive exhibits. Personally I wouldn't waste the money, but it's there if you want it.
Fun Alternatives: If you want to experience great views of the city for free, go to Gasworks Park, Volunteer Park or Kerry Park. If you are looking for family friendly things to do that are actually worth the admission, try the Seattle Aquarium, Woodland Park Zoo or Seattle Art Museum. See my tips for more info.