jis is correct.
Let me add that, to get from King Street Station to the International District Station, exit the main door on the west side of King Street Station, turn left (south) and walk to the Weller Street footbridge, and turn left (east) and take the bridge. You have your choice of broad stairs or an elevator to get up to the footbridge. Take the bridge over the railroad tracks to 4th Avenue S. Cross 4th Avenue S at the crosswalk, keep going straight (east) through the pedestrian plaza between the two buildings, and the entrances to the International District Station will be on your left (north). To get on a southbound Link light rail train, you will want to take one of the entrances on the west side, to Bays C and D. (Once you are down on the platform, it won't matter whether you are at Bay C or Bay D, as the trains are long enough you can board from either.)
Here is a map/diagram of the International District Station, and the construction company that built the footbridge has
some nice pictures of it.
The Weller Street footbridge is also the way one would get from King Street Station to the Sounder (heavy rail commuter) platforms.
The footbridge is your best bet, but, alternatively, you could walk straight (west) out the door of King Street Station, go halfway up the block on S King Street, take the curving staircase attached to the building on your right (north), walk north on the sidewalk at the east side of the building until you come to S Jackson Street, turn right (east) on S Jackson Street, cross 2nd Avenue Extension S and 4th Avenue S, and after you pass the front of the old Union Station, the entrances will be on your right (south). Or if you didn't want to wrestle your luggage up a staircase, you could walk S King Street to 3nd Avenue S, turn right (north) and take it one block to S Jackson Street, turn right (east) on S Jackson Street, and continue as previously stated.