Horseshoe Meadows would be a great place for you and your group to acclimate and great for a day hike. You could hike up to Cottonwood Pass or even up to Trail Peak. Not too far from Whitney portal. There are many posts on here regarding horseshoe meadows. Make sure you find out an=bout road conditions first before going up.
As for the crampons and ice axe, I would advise to way more practice with both before you go up the MR. It's a steep ascent and can be icy in parts - depending on snow conditions in June.
