Any member of the legislature can introduce a bill. There is no limit to the number of bills and resolutions a member can introduce. However, there are limits to the number of co-authors. In the House, there can be 35 total authors; in the Senate, the limit is five. Once a bill is introduced in either body, the chief author may find someone to carry the companion bill in the other body. A companion bill is usually identical when introduced, though each body's bill may change independently after introduction. In the House, the Speaker of the House assigns each bill and numbered resolution to one of the standing committees. The Chief Clerk of the House then assigns each bill a House File number, which will identify the bill in its travels. The Senate has traditionally used a somewhat different path to introduction. Bills and resolutions are given a number by the Secretary of the Senate's Office and assigned to a committee by the Senate President.
Once introduced, a bill travels through the committee process. It goes through the relevant policy committees and, if it has financial implications, a finance committee. After committee discussion, members can recommend action. Typically, the bill is tabled, laid over for inclusion in an omnibus bill, or sent to the floor. A bill must receive three readings on the floor before members debate it and take a final vote. Many bills never make it that far. To get a sense for how many bills are introduced and passed, please see the Legislative Reference Library's page: Number of Bills Introduced and Laws Passed in the Minnesota Legislature, 1849-present.
Once a bill is passed by the House or Senate, the bill must travel to the other body. If the two bodies do not agree, the bill will travel to a conference committee. See the FAQ: What happens when a bill has passed one body but not the other? for more information about conference committees. Once the bill is approved by both bodies, it goes to the governor for his approval. If the governor signs the bill, it becomes law.
See the steps a bill takes to become law on the page: How a bill becomes law in Minnesota, which combines information from the various offices of the Minnesota Legislature on the process of lawmaking in Minnesota.