Most of the high stuff will be closed until mid-late June. They do run bulldozers through the backcountry trails to open them up earlier than simply letting them melt out because of the tourist jeep trade, but it's still way early for that.
I would head into Ouray, park, and walk down the street on the west side until you find an outdoor shop that sells maps, etc. Ask them what's open. Alternatively, ask one of the Jeep rental places - they should be able to help you.
You might be able to get partway up into Yankee Boy Basin. Also, the Last Dollar Road between Ridgeway and Telluride should be pretty - not a real "trail", however. The road over Owl Creek pass north of Ridgeway to Silver Jack reservoir might be open as well.
