You already have a good basis for demonstrating your cross-cultural skills: studying abroad for a semester in Italy and a month of travel in Europe. You will also be working in a field (computer science) with a strong track record for international employment. The most important thing you can do now is target the type of international employment you are looking for, and then to seek professionals currently working abroad in your field. Ask their advice and tap into their knowledge of the international aspects of your field. For example, you may target short-term work in Ireland or permanent residency in Australia; both of these countries have a tradition of hiring computer specialists. Similarly, you may decide to target working for an NGO or international organization. Each field requires a different set of approaches for landing an international position.
You've given yourself a workable deadline for this goal. There are a number of things you can do over the next five years to continue building your international credentials. Look for international opportunities at your current place of employment by offering to work on international projects and/or with international work teams. Get experience by managing an outsourcing project (in South-East Asia, for example) and learn to manage a project across cultures and time zones. Join international umbrella organizations in your field of expertise. Start acclimatizing your family to overseas living by going on one or more cross-cultural vacations (if possible, live with a local family and not in a resort), and begin learning the native language/s of your targeted locations. You might also consider going on a job exploration vacation and starting to network internationally.
The key to all these activities is to position yourself as someone who is skilled in an international work environment. The more experiences you accrue, the better you will be able to demonstrate that you can easily integrate into a cross-cultural work environment.
In short, start learning how others in your field are working internationally and network with this group while continuing to build your international skill set. Don’t wait too long to get started. Your family is still young, and its tolerance for risk-taking (i.e. moving abroad) is greater today than it may be in later years when you find yourself settled with mortgage payments and children enrolled in school. It is ideal to take children abroad when they are between the ages of three and 13. It is my belief that high school students should be schooled at home with a close, stable network of friends, unless they already have a long history of living overseas. Family life abroad is often rich and full of meaningful family-centered experiences, so take advantage of the opportunity to head abroad sooner rather than later.
All the best to you and your family as you set out on your international adventures! Also check out Personal Relationships Abroad (for registered users).