Big question.
Political: the democracy, the republic. Having representatives for groups of people. Having a strong military with extra perks for service like high pay, and land to retire on.
Trying to set up a global market. They were big on luxury, arts and exotic foods. Just like we eat oranges and grapes in cold areas or import Egyptian cotton and silk.
Social is pretty hard to peg down since there is so much that follows with the political styles. There was also how property and family lines were carried on. The man's name going to the children, etc.
They prized aggressive sports that showed strength like gladiator events, the olympics, races. We have football, the olympics and have upgraded our horsed into race cars.
They were pagan which influenced a lot of our literature even after the majority people in that area switched to christianity. Shakespeare is a perfect example of that. Also movies like Hercules from Disney. They have become part of our entertainment. Rome from HBO...
Take a look at architecture too. The white house, the eagle (used by us and the nazis), columns on all our official buildings. We use marble on all our government buildings too. Its not because we have a ton of it, its because the originals were marble.
