What sort of activities are you looking to do? If just driving around, that's one thing. But if you're a big hiking or camping fan, that changes things a lot. Fishing, sightseeing, and whale watching are common things to do. How many are in your party, any kids, any mobility issues? Is photography or shopping a major activity? Are you a five-star-spoil-me kind of visitor, or are you more independent, choosing to make your own fun?
Washington State has two sides to it. Western Washington is greener, has the water, and the big Parks. Eastern Washington is high desert, more open, ranging territory, and has some amazing places to see. Weather in early June will likely be good (for us), but not hot. It is rarely above 70 degrees in June in Western Washington, and it can rain at any time. Plan your trip around the activities you want to do, and dress in layers - if it gets warmer, take more off. But be ready for things to cool off as the sun goes down.
Drive times can affect what you want to do, and the amount of time you'll have to do it. Traffic around the major cities, especially Seattle and Tacoma, can be brutal, as there aren't many highways, and only two primary Interstate freeways - one that runs north and south, the other from Seattle going east.
Please offer more information, and I'm sure you'll get lots of help.
Dave