Here's how you could think of it:
* The Western "Start": The Strait of Gibraltar is the narrowest point between the Mediterranean Sea and the Atlantic Ocean. So, you could say the Mediterranean "starts" there.
* The Eastern "Start": The Mediterranean Sea stretches eastward, with countries like Turkey, Syria, Lebanon, Israel, and Egypt bordering its eastern edge. You could consider any of these countries to be on its eastern "start".
Ultimately, it depends on what you consider the "start" to mean. It's a sea that is connected to the Atlantic Ocean, and has a very diverse coastline with many countries surrounding it.