One significant difference between both parties existed in the role of national government. Federalists advanced a strong and centralized role of government. This was rooted in their belief that the "best" and "brightest" should lead the nation. Since these individuals would most likely be in the position of governmental power, then it was acceptable for government to be an expansive entity. The Democratic- Republican party featured a lesser role of the federal government, opting for local governments to exert a greater role in citizens' affairs. This was rooted in the constituency of the Democratic- Republicans. Farmers and people who were of more common extraction comprised the Democratic- Republican party, while the Federalists were filled with more wealthy individuals and people who were successful in business and economic affairs.
From this point, most differences towards the national government were evident. A "strict" interpretation of the Constitution that the Democratic- Republicans featured was posited against the "looser" interpretation of the Federalists. The Federalist desire to enact a national tariff was countered with the opposition of the Democratic- Republicans. The attitudes towards the national government that both parties demonstrated reflected fundamentally different and theoretical starting points.
Jefferson was the leader of the Democratic-Republican Party (not to be confused with today's Democratic or Republican parties). His mantra could be described as "power to the people". He was in favor of a small central government with individuals retaining as many rights as possible. He thought that the states should be stronger than the federal government. Jefferson also thought that agriculture should be the backbone of American society. His views were more compatible with small-town farmers and plantation owners, particularly in the South, than wealthier businessmen.
Hamilton was the leader of the Federalist Party. The Federalist Party was supported by bankers, doctors, lawyers, and other wealthier landowners because Hamilton was in favor of making up the ruling class from the wealthy. These wealthy would make important decisions and run the government. Thus, Hamilton believed that a strong central government would benefit the United States by allowing wealthy, successful businessmen and landowners to control the nation. Most of Hamilton's supporters were in New England. Hamilton's mantra could loosely be described as, "power to the rich".
This is a great question. First and foremost it is important to remember that the term "Anti-Federalist" is misleading. The term gives the impression that there was a political party, like today's Republican or Democratic Parties, called the Anti-Federalists. In fact, there was not. Historians use the term "anti-Federalists" to describe, or lump together, all the people from across the newly independent United States who were against the ratification of the Constitution. People opposed the ratification of the Constitution for many different reasons, and these people came from many walks of life, and lived in many states and other than being opposed to ratification of the Constitution, they had little in common and it does them an injustice to give them one simple "anti-Federalist label."
Believe it or not both of these groups had a major idea in common. Leaders of both groups shared a belief about the national government, that it should not be to strong or centralized and that the system of division of power or Federalism leaves most power in the state. Both feared a national government that would become too strong. Both groups believed that state and local governments would do a better job representing the beliefs of the people. Both sides feared a central government that would be overreaching and too powerful.
Ok, to answer your questions directly, while both sides feared a central government that was too strong, the Federalists supported the ratification of the Constitution and the creation of a stronger Federal government, a stronger government than that existed under the Articles of Confederation. The anti-Federalists felt the Constitution as written took too much power from the states and also felt the Constitution did not protect individual rights of citizens enough from the power of this new stronger central government. That is why, A Bill of Rights, the first 10 Amendments of the Constitution will be added. Thanks to the Anti-Federalists, in exchange for their eventual support for ratification, a Bill of Rights was added to the Constitution.
For a more detailed look at this I would suggest reading some of the Federalists Papers written by Hamilton, Madison, and Jay and compare them to the works of anti-Federalists such as Patrick Henry. Or a great book to check out is "Ratification" by Pauline Maier.
One final word, please don't mix up the differences between the Federalists and Anti-Federalists and the later split between the Federalist Party and the Democratic-Republicans, that is a different subject.