Change in church culture requires 18-24 months, so aim at several smaller objectives on your way into the global community. First, target one particular portion of the world. "Go into all the world" is not instruction to a single congregation, but to Jesus' entire church. Congregations actually diffuse their global awareness and impact when they scatter their energies across several countries/continents. Engage your congregation in the process of picking where to focus your attention, and they will already have increased interest in the world.
Next, solicit contact information about people/ministries in that country/region. This is another great way to involve the congregation. Who has ties to or a burden for that part of the world? Who would like to create a History Channel-type presentation about it? Does anyone speak a language used there? Closely related to this step is the need to build a global view on relationships. You will get more buy-in from your congregation if you introduce them to specific pastors, orphanage directors, and ministry leaders. God wired us to give to others, not necessarily to causes. Present the world to them through "so-and-so who has labored for years in …"
The best way to do that is another of your key objectives: get members of your congregation (especially yourself) to your target locale. Other parts of the world are like different kinds of food. Almost invariably, your people will fall in love with wherever you take them. Nothing quite tops the experience of being in a vastly different setting and being used by God, in even the smallest way, to touch the world.
Initially, the team should be very small; its mission should simply be to look around, meet individuals and encourage them (i.e., be like Jesus). As the team debriefs, it will definitely get a sense from the Lord about what your church can—and cannot—do. On the trip back home, encourage each team member to recall five specific stories to tell everyone back home. Informally scattering those travel accounts among the congregation will do more to change your church culture than a dozen sermons.
I conduct pastors/leaders' seminars in over 30 nations each year. The single biggest need I observe is a simple one for any local church to provide: ours is a lonely business, and ministry leaders everywhere crave relationship with other leaders. Similarly, congregations throughout the world love lasting connections with other-country cousins.