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.