Getting from Seatac Airport to the Seattle Center area will most likely require a taxi (approx. $35) or shuttle bus (approx. $40). (Our light rail should be in place for 2009.) There is public transportation (approx. $2), but it is much more time consuming. Here is a map to help.
Once you are in Seattle proper, transportation should be relatively easy. The bus system is fairly comprehensive. There is a "Ride Free Zone" within downtown Seattle (The Seattle Center is a bit north of "downtown", and a bit outside the zone - but within walking distance of the zone). Also, there is the famous Monorail, which will get you from the Center to Downtown and back ($4 round trip, $2 one way).
Travel tips and information about Seattle.