Question

what kind of court cases start at the supreme court level

Answer 1

According to Article III, Section 2 of the Constitution, two types of cases may begin in the US Supreme Court. This mean the Court has original jurisdiction and may hold a trial, if necessary.
affecting ambassadors and other public ministers and consuls

disputes between the states (original and exclusive jurisdiction, see 28 U.S.C. § 1251)

The Court only has original and exclusive jurisdiction over disputes between the states, however, per 28 USC § 1251. The Supreme Court has concurrent (shared) jurisdiction with the US District Courts over cases involving ambassadors and other foreign dignitaries. The District Courts typically hear these cases first.

In all other cases the Supreme Court has appellate jurisdiction, except where Congress has explicitly stripped them of jurisdiction.

Title 28 U.S.C. § 1251

Original Jurisdiction

(a) The Supreme Court shall have original and exclusive jurisdiction of all controversies between two or more States.

(b) The Supreme Court shall have original but not exclusive jurisdiction of:

(1) All actions or proceedings to which ambassadors, other public ministers, consuls, or vice consuls of foreign states are parties;

(2) All controversies between the United States and a State;

(3) All actions or proceedings by a State against the citizens of another State or against aliens.