The graphic From Bill to Law shows the stages of the legislative process.
The Congressional Research Service report Introduction to the Legislative Process in the U.S. Congress contains a detailed description of the process, by which a bill becomes a law. The summary here only considers the House actions.
Introduction and Referral of Legislation. Once a Member of the House ... introduces a bill, it is typically referred to the committee (or committees) in that chamber with jurisdiction over its elements. Committees do not formally consider each of these referred bills. The committee chair has the primary agenda-setting authority for each committee and identifies which bills will receive formal committee attention during the course of the two-year Congress. (Committees are not limited, however, to consideration of measures referred to them and may initiate legislative action on their own.)
Committee Consideration. A committee may conduct hearings on a bill to provide committee members and the public an opportunity to hear from selected parties (e.g., a federal agency or organized interest) about the bill’s strengths and weaknesses. If the committee wants to formally recommend that the bill receive consideration from its parent chamber, it will hold a markup on the bill, at which committee members vote on any proposed amendments. The markup concludes when the committee agrees, by majority vote, to report the bill (with any recommended changes
adopted in the markup) to its chamber.
Floor Scheduling. In the House, majority party leaders generally decide which bills will receive floor consideration; typically, they schedule a bill for a type of streamlined floor consideration, or instead ask the Rules Committee to propose a set of tailored parameters for floor consideration. ...
House Floor Consideration. In the House, most bills that receive consideration do so under a procedure called “suspension of the rules,” which limits debate to 40 minutes and prohibits floor amendments, but requires two-thirds of Members voting to agree. Most other bills are considered under tailored debate and amending parameters set by the terms of a special rule reported by the House Rules Committee (which effectively operates as an arm of the majority party leadership). The House first votes to adopt the special rule, and then can proceed to debate and potentially amend the bill (typically accomplished in a setting called Committee of the Whole). After any debate and amending process is complete, the House then typically votes on a minority party alternative (through a vote on a motion to recommit) before proceeding to a final vote on passage.
The Speaker's responsibility is to direct new bills to the appropriate committee(s) and is defined in Rule XII – Receipt and Referral of Measures and Matters.
If or when the bill is returned from the committee(s), it is placed on a calendar. During consideration of legislation, the Speaker (or pro tempore), in Rule XIV – Order and Priority of Business. "shall call each standing committee in regular order and then select committees. Each committee when named may call up for consideration a bill or resolution reported by it on a previous day and on the House Calendar. If the Speaker does not complete the call of the committees before the House passes to other business, the next call shall resume at the point it left off, giving preference to the last bill or resolution under consideration."
When the Committee of the Whole House debates and votes on a particular bill, the Speaker (or pro tempore) is responsible for controlling the debate in accordance to the rules.
While the Speaker does far more than this brief description, I have limited the description to the movement of a bill from introduction to vote.
However, most of the power is exercised by majority members rather than the Speaker. A committee chair may decide to not consider a bill referred to it. Even when reported by a committee and on the calendar, the majority leader may choose to not bring the bill to the floor for consideration.