To elevate or improve one's social, political, and/or financial position in life; to become more successful than one was before. You're only going to truly rise up in the world if you make a point of rubbing elbows with those of a higher social standing. It's unsurprising how quickly Sarah has risen in the world when you consider that her tenacity and determination are only matched by her intelligence and talent. The Robinsons really rose up in the world after they won the lottery.