There isn't officially a first Prime Minister, since the post wasn't officially created but evolved gradually - it used to mean just the first or chief of the king's ministers.

However, the nearest thing to a modern day Prime Minister was probably the office held by Sir Robert Walpole. Walpole was a prominent Whig politician who found favour with King George I, largely because George was unpopular with the only other political party of the time, the Tories. After a financial disaster in 1720 (a bit similar to the one in our own times) George reapponted Walpole First Lord of the Treasury in 1721. He governed specifically through the House of Commons, increasing its importance and power relative to the House of Lords. By this time it was clear that the days of 'the divine right of kings' were over, eg the king was more of a symbolic figure and real power should reside with elected representatives. So although Walpole always referred to himself as the king's servant, he was effectively in charge of running the country.

Walpole was also the first leader to take 10 Downing Street as his official residence.