Let's begin with formal powers and duties. Of the President, Article II Section 3 of the Constitution says:
He shall from time to time give to the Congress Information on the State of the Union, and recommend to their Consideration such Measures as he shall judge necessary and expedient; he may, on extraordinary Occasions, convene both Houses, or either of them, and in Case of Disagreement between them, with Respect to the Time of Adjournment, he may adjourn them to such Time as he shall think proper; he shall receive Ambassadors and other public Ministers; he shall take Care that the Laws be faithfully executed, and shall Commission all the Officers of the United States.
By convention, the President fulfils the first bolded clause with an annual speech called the "State of the Union Address." Normally, this will cover the economy, other domestic (and, increasingly, foreign) political issues, the government's budget, and any major policy initiatives which the President is currently backing. The other party will usually give a rebuttal speech immediately afterwards, which gets far less media coverage and is generally thought of as uninteresting. Unlike the first speech, the main goal of the rebuttal speech is to get through it without making any gaffes. Third parties sometimes also make rebuttal speeches, but these are widely ignored.
The second bolded clause imposes a separate duty on the President to make recommendations to Congress. However, it is a generalized sort of duty, which does not need to be fulfilled at any particular place or time, nor with any particular frequency. This is also true of the State of the Union Address, which is only held annually by convention, not by legal requirement.
(It may seem surprising that the Constitution imposes duties on the President and then fails to specify how and when they should be performed. It must be stressed that each branch of government is independent of the others. Congress or the Supreme Court cannot compel the President to make recommendations, because that would impinge on the separation of powers. The Recommendation Clause should be thought of as more of a "civic duty" than a legal responsibility.)
The third bolded clause has not been needed since 1948. When the Constitution was written, a continuously operating Congress was thought impractical, so Article I allows Congress to recess as it sees fit, provided the Houses can agree on the times and dates of any recess longer than three days. But in the event of a national emergency, it might be necessary to call Congress into extraordinary session, cutting the recess short. The Constitution gives this power to the President. Nowadays, Congress generally meets year-round with a few short recesses for holidays, so an extraordinary session is much less likely to be necessary.
Moving on to shared powers, Section 2 of the same article provides:
He shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur; and he shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law: but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments.
Legally, the Senate may provide "advice and consent" by a simple majority (or a two-thirds supermajority in the case of treaties). This takes the form of a "motion to advise and consent." The Senate need not actually provide any material "advice" beyond its assent, but presidents often do have informal discussions with individual senators (to determine that senator's opinion of the proposal) and with their party leadership (to determine whether sufficient votes exist). Of course, the President will also confer with party leadership and individual members of Congress about regular legislation in much the same fashion, for largely the same reasons.
Finally, moving on to conventions, there really aren't any. The President will often meet with both parties' leadership if a bill needs to pass (or fail) in Congress, if the President wants to exercise one of the powers listed above, or sometimes just because. There are, however, several bills that must pass every year (or every N years) in order to keep the government functioning normally:
If one of these bills is unable to pass, the President cannot reasonably ignore it and must work with Congress. Otherwise, one or more critical functions of the federal government would have to cease due to a lack of authorization or under the Antideficiency Act (which requires a government shutdown in the event of a lapsed budget).