How much power does the position of US president really have? Is he not, to a large extent, dependent on advisors etc who could manipulate quite easily?
Only to the extent that he lets himself be (no President in particular being referenced here).
The (generic) President, in propria persona, has the final authority on anything which is Constitutionally or statutorily his prerogative. To the extent that he does not draw on the expertise of his expert advisors, he’s being a fool. To the extent that he lets them dictate his responses, he’s abdicating his own responsibility.
Roosevelt and Truman both had to assert the authority of their office to quell subordinates who had an overly high opinion of their own authority. (Source: Manchester’s The Glory and the Dream and McCullough’s Truman)
Mr. Truman, with no strong background in foreign policy, placed a lot of trust in what Gen. George C. Marshall and Dean Acheson advised him – but he was clear that he and he alone would make the final decisions, and both men respected him enough not to argue with that.
Another thing to keep in mind is that the US President is both the head of government and the head of state. He is also the commander in chief of the armed forces. (Source: CIA World Fact Book)
You don’t need the CIA to tell you that. 
The President’s powers are laid out in the Constitution, which in most places is quite concise and fairly straightforward. Highly reccomended reading.
Congress may also grant the President executive authority through legislation. For example, the executive departments (and the appointed positions therein) are created by congressional legislation. The same legislation says the President is the boss of those departments.
The modern presidency operates in a mesh of constraints.
Legal constraints includes the checks and balances powers of Congress and the Supreme Court.
Operational constraints include the lack of time to fully absorb and understand the implications of every issue on earth, the limited amount of understanding that others have or can gather on those topics, and the competing power groups inside the administration that offer differing advice.
Economic constraints include the budget that the president has proposed and Congress passed, the lag time between proposing a plan and implementing it, the difficulty of reallocating funds, and the country’s financial well-being.
Social constraints include the reactions from other politicians, other countries, the media, the general public, and the president’s family on what is acceptable behavior for a president, before and after the fact.
Institutional restraints include the independent powers, customs, histories, and cultures of the military, the spy organization, the cabinet departments, the independent agencies, the Federal Reserve, and all the other arms and branches of government.
Personal restraints include the reactions of the president’s advisers to the president as a flawed human being. An example recently revealed is this one.
Other than that, the president has awesome powers.
If the President were to just train enough he would have plenty of real power.