Mono Lake itself, eerie in its stillness, is the main attraction to this remote area of the Eastern Sierra. It might be enough just to sit on the edge of the lake and enjoy its beauty. Or in the summertime, enjoy an oddly buoyant swim in the heavily salted waters or even a boat trip  out to some of the silent uninhabited islands.
But if you’d prefer more adventure or exploration , a number of options await.
Getting to Mono Lake from San Francisco  or Los Angeles  or anyplace else in California with a major airport is, frankly, a royal pain in the you-know-what. Mono Lake sits almost on top of the junction between Tioga Pass Road and Highway 395.
Tioga Pass  is closed between November and May each year—check with the Yosemite website  to get the exact closing and opening dates for this road. Highway 395 remains open all year (though storms can close it briefly until it is plowed). But accessing Highway 395 from the north or south involves long drives. You might want to consider flying in to Las Vegas or even Reno to get to Mono Lake as directly as possible.
Very little public transit of any kind gets as far as Lee Vining and Mono Lake. To adequately explore this region, you need a vehicle of your own. On the bright side, parking in Lee Vining and around the lake tends to be both easy and free. Few enough others are vying for spots that you won’t spend much time looking for your own space.