If your starting point is London, your best destination point is Rome. From there, you can travel to southern Italy to hit the beaches. That's the best place, since Greece is a mess, and not only with regard to transportation, and the former Yugoslavia is not much better off.
There are two basic routes to get to Rome: the French route, and a German route.
Taking the French route, your main stop would be Paris. Crossing the North Sea, you might first want to head for Antwerp and Brussels before heading south. The sights of Paris are legendary, the Eiffel Tower, the Louvre, the Champs Elysees, and the Etoile. Plan on spending several days there, then head east through the wine country. From there, you may want to travel down the River Rhone to northern Italy, if you prefer "water," or through Switzerland, which has the advantage of the natural beauty of the Alps.
The other route is through Germany, perhaps via Amsterdam, and then Berlin, which features old sites such as the AlexanderPlatz and the Unter den Linden Boulevard. Heading south, you might want to see the city of Dresden, known for its "china" (porcelain). You might cut across the Czech Republic and take in Prague, with its churches and monastaries, before seeing the Austrian Alps, arguably the most beautiful of the four (French, Swiss, and Italian are the other ones). The other advantage of this route is that you will be able to eat more heartily and cheaper, at least in the former East Germany.
Both routes converge in northern Italy. You'll probably want to see the art museums, plus Michaelangelo's sculpture of David in Florence. From there, it's onto Rome for the Colosseum, the Spanish Steps, and the Wedding Cake monument. For some good eating and a view of 500 fountains, there is Tivoli, nearby. Then to the beaches.
Have a safe trip.