Frank, Mt. Dana is a good one, as is Mono Pass. Both are short hikes, and Mt. Dana will get you to 13K' and back in half a day. We'll be out there in less than 3 weeks, sleeping 5 nights at Mammoth Lakes while dayhiking in Yosemite. Cathedral Peak and Cloud's Rest this time around. You can't beat sleeping in a Mammoth condo at 8K' with a blazing fireplace after a day of hiking in Yosemite. This will be our 3rd trip doing just that, staying in the same condo. After 5 nights in Mammoth, we'll shoot down to Lone Pine and plan to do the Whitney MR and Langley - weather, permits and legs allowing.
A quick side-trip in Yosemite is Olmstead Point. Huge parking lot and an overlook, with a granite dome behind it. Lots of people climb up this dome about half a mile for better views, but if you keep going you'll negotiate a small saddle to an even higher point. It may take some scouting to find your way up, but it's worth the effort. From this vantage you can see not only the valley to the west, but far back to the east including Tenaya Lake. I promise you'll be alone.
There are a couple of trails at the Ancient Bristlecone. We've hiked the Methuselah Trail, and it's well worth it. Short, but outstanding views and tons of Bristlecone pines. The drive up to the trailhead is fun too.
That BBQ restaurant you mentioned may be the Whoa Nellie Deli, an iconic eatery located in the Mobil station at 395 and Tioga Rd (120). Gourmet meals at a gas station - only in California. They have quite a menu that does include some BBQ, but also a variety of other items. The lobster taquitos are my favorite. Big green lawn with picnic tables overlooking Mono Lake, and often live entertainment. A must-experience if you've never been there.
