Do interest rates fall when the money supply rises, because there is excess reserve and the banks have to loan it out to someone to make a profit?
So they lower the interest rates to give people an incentive to take loans. So there is surplus of reserves and whenever there is a surplus, the price of anything goes down. In this case, the interest rate. Is the above information entirely correct?
Yes, that is correct. The interest rate is the price of borrowing money. The money supply is the supply of money to be borrowed. As you will know since you have taken basic economics, when the supply of a good or service goes up, there is a temporary surplus of that good. When there is a surplus of a good or service, the price of that good or service must go down because there is more of it than people want at the previous equilibrium price.
So, your analysis is exactly correct. The money supply goes up causing a surplus of money. That makes the price of borrowing money (interest rate) go down.
We can thing of interest as cost of borrowing money. and Money supply as the money that is available for spending directly or through borrowing. Thus when money supply increases, the money available with banks for lending increases, and in line with the law of demand and supply, the banks need to lower the interest rate to lend the additional money available with them for lending.
Also, the banks also try to reduce the interest paid by them on the money deposited in the banks by people when they have more money than the they need for borrowing at the original interest rates. Thus interest rates paid by the bank as well as that charged by the bank tend to fall when the money supply increases.