From the West, use the Jewel or Ammonoosuc Ravine trials. These will get you to the 'Lake of the Clouds' AMC hut (reservations needed for overnight stays and hot dinners). Hot showers may be available. These trails are more gradual and wooded initially, but can be congested on weekends. From the hut you can climb to the Summit or hike along the Great Gulf Trail to summit the other presidential peaks. Be aware that the weather can changed within minutes and you can die from hypothermia even in the summer. Mt. Washington has killed as many people as Mt. Everest.
From the east, use the Tuckerman Ravine trail; its shorter but some sections are very steep, but offers good views. Descend via the Lions Head trail back to base camp (AMC Pinkham Notch hut) which offers lodging, hot showers, and excellent meals, especially dinners served community style. Make sure you and your companions sit at the start (beginning) end of the bench where you can serve yourself the best portions.
There is a visitor center at the Washington summit where you will run into fat ladies in high heels who ride up via the cog railway, take mini-buses, or private cars. After summiting many times, I no longer go to the top (been there, done that), but instead just enjoy the scenery below the summit. The Alpine Garden just below the summit should be in full bloom in early June. As noted before, prepare for the worse (flashlights, layers of warm clothes, wind breakers, etc.)