Well, whenever a volcano erupts, there is usually a measurable decrease in average temperature because of all the ash and gases in the atmosphere. Assuming that Yellowstone is one that erupts properly instead of just crumbles, it probably would drop temp significantly (possible into the next ice age).
Assume that most of the US and Canada and Mexico are blown away. This would cause tsunamis to slam into every coastline that you could you project a relatively straight line from the explosion. I'd probably say it would be coming at a couple hundred miles an hour and would wipe out everything from the shore to the mountain. It would also kick up a lot of stuff around, mixing up surface dirt, exposing other places to the bedrock, but also spreading the coastline out when the water drags everything with it as it recedes. That means you'll find buildings, artifacts, and stuff really far into the ocean.
Obviously, this will change global weather patterns because the Rocky Mountains wouldn't be there. That would cause changes to the air flow, weather, and precipitation. When the lava cools, it probably would create a new mountain range of rock, which means there would be a wasteland for quite a while.
Once the glaciers from the new ice age recede, there is probably going to be a new sea where Yellowstone was. It probably will also join in with the coastal ridge so you'd probably have the plate tectonics spreading away from that point instead of contracting toward it.
Having a big crater will also lower the ocean levels relatively quickly. How much? Not entirely sure. Some of it will be deposited on the existing continent that wasn't destroyed while more of it would return. I'd probably expected a few inches lower at least.