What issues with the original Constitution led to later amendments?

In an effort to preserve the relevance of the government for future generations, the founding fathers made the constitution a living document. This means that the form and function of the government can be expanded and changed through the process of adding amendments. Many of these amendments addressed specific social failures or design flaws that were either not understood at the time or became more relevant as time went on.

The first ten amendments, or Bill of Rights, addressed the criticism that the rights of the individual were not protected from the power of the government. They protected specific enumerated freedoms and divided power between the federal and state governments more.

Amendment 12 was aimed at reworking the presidential election system to address the development of political parties and make it less likely that members of opposing parties will share power in the executive branch.

The 13th, 14th and 15th Amendments passed after the Civil War sought to extend civil rights to former slaves. They tried to make it less likely that African Americans in the south will be socially equal. The 19th amendment extended this right to women in the 1920’s.

The 17th Amendments expanded the principal of popular sovereignty by allowing the people to elect their state senators rather than them being appointed by the states themselves. This was further expanded by the 26th amendment, which extended voting rights to all 18 year olds.       

Sometimes amednments correct problems with other amendments. In a strange “double-wammy” decision, the 18th amendment tried to ban alcohol, but the 21st amendment revoked the 18th went it was clear that the American people weren’t willing to give up their alcohol.

The first goal in the Preamble to the Constitution is the idea of remaining committed to "forming a more perfect union."  This fundamental goal suggests that the Constitution is a document that can respond to the demands and needs of its particular time period.  In assessing that American democracy is an ongoing experiment, the Constitution presented itself as being able to show strength through pliability in the conditions that would be featured in American society.  The results of the Civil War in which the North had defeated the slave- owning South was one of those conditions. Specific changes that needed to be enshrined in the Constitution resulted in the ratification in the 13, 14th, and 15th Amendments.  The suffragist movement in America that spanned over seven decades did much to cause change in the way people viewed women, and thus pass the 19th Amendment.   American society endured another change in thought when it came to examining the role that young people played in society.  While eighteen year olds were able to fight and die in the Vietnam War, the argument became accepted that they should be able to to vote for the leaders that might choose to commit them to such entanglements.  In this, the Constitution reflected social changes in thought through the adoption of the 26th  Amendment.  In these instances, previously marginalized groups helped to change the thinking of American society.  These changes resulted in new amendments being adopted in the hopes of "forming a more perfect union."

What issues with the original document led to later amendments?

I am assuming you are referring to the amendments after the Bill of Rights. There were many problems and societal forces that required changes. I will discuss just two.

One set of circumstances that led to the 14th and 15th Amendments was the aftermath of the Civil War.  Bear in mind that the Bill of Rights guarantees that the federal government must ensure the rights contained therein.  There is nothing in the Constitution or the Bill of Rights that promises the states will preserve any rights at all.  After the Civil War, it became apparent that the Southern states were going to do everything possible to make life difficult for the slaves who were now freed.  So the 14th Amendment makes clear in its language that the states must provide all the "privileges or immunities" of citizenship, and the 15th makes clear that race may not affect these rights. Without this language, the states would have been free to do as they pleased.

An interesting circumstance led to the 22nd Amendment, the administration of Franklin Delano Roosevelt.  Roosevelt was elected to office four times!  There was a feeling that this was too long for anyone to be president, so the 22nd imposed term limits on the presidency.  Roosevelt had died before the ratification in 1951, so he had no opportunity to be elected a fifth time, but the principle is a good one.  Too many years of one person at the helm is probably not a good idea.

For most of the amendments following the Bill of Rights, there is a story. What you might notice as you examine them is that in many cases they were reactive, rather than proactive, ways of making adjustments to situations the framers had not contemplated or responding to changes in society.  Even today, most of the discussion about any new amendments is quite reactive, for example, the discussion about an amendment to require marriage be only between a male and female.  This is clearly a reaction to the assertion of equal rights by gay people, creating a backlash among various groups who believe that homosexuality is "wrong."

What societal changes and problems with the original constitution motivated amendments?

I assume you mean amendments other than the original ten, as they were passed shortly after the adoption of the Constitution. It is impossible to create a laundry list of each amendment; but a few notable examples are in order:

The thirteenth, fourteenth and fifteenth amendments were passed at the time of the civil war. They were designed to end slavery, guarantee citizenship and equal rights to the freed slaves, and guarantee that they had the right to vote. As originally drafted, the Constitution was silent on the issue of slavery, although its language led most scholars to believe that it was constitutionally protected. For that reason, a constitutional amendment was needed to end it. By the same token a constitutional amendment was needed to protect freed slaves.

The Sixteenth Amendment provides for an Income Tax that need not be apportioned among the several states as provided in the original constitution. The Seventeenth provides for direct election of Senators; and the Eighteenth was the "prohibition" amendment. These were all part of the Progressive movement of the late nineteenth and early twentieth century in which it was believed that the cure for the ills of democracy was more democracy. The eighteenth was repealed by the 21st Amendment which had proved unworkable.

The twentieth amendment changed the dates for Presidential inaugurations. With new methods of tabulating and reporting votes, the long wait which had hitherto existed were no longer necessary; plus the long "interregnum of despair" between Franklin Roosevelt's election and inauguration indicated that a long lame duck term was not good.

The twenty second amendment limited the President to two terms. This was after Roosevelt had been elected to four terms; and it was generally agreed this had been a mistake. Roosevelt had been very ill by his fourth term and in fact died in office. He made concessions at the Yalta Conference which a stronger, healthier president might not have made.

The twenty fourth amendment outlawed poll taxes, a device used in many Southern states to disenfranchise black voters. It was part of the growing civil rights movement of the time. Finally, the twenty fifth, which provides for Presidential disability and succession, including the power of the President to name a new Vice President, was passed after the death of John F. Kennedy. He was succeeded by Lyndon Johnson who had previously suffered a serious heart attack. If Johnson had died in office, the next in line to the Presidency would have been Strom Thurmond, president pro tem of the Senate who was also up in age, and then the Speaker of the House who also was elderly. It was felt that a more orderly succession needed to be provided for.