The biggest problems facing the global economy (starting with the biggest) are:
1) Lack of globalization in most of Africa, Latin America, and the Middle East. Fortunately, most of Asia has finally caught on that this is a great thing, and their economies are booming as a result. How do we fix this problem in those other regions? First of all, in countries that don't already have democratic traditions (most of Africa and the Middle East), there needs to be a strongman dictator in favor of neoliberalism. After the standard of living subsequently improves, people will be more aware of the outside world and more able to handle democracy. In the countries that DO have democratic traditions (most of Latin America), there's really not a whole lot that can be done other than allow them to watch the neoliberal countries around them boom while they continue to wallow in socialism.
2) Global dependence on oil. Not only is it polluting our planet and warming it at an alarming rate, but it funds terrorists and nasty dictatorships. We need TEMPORARY government subsidies (did I mention that they should only be temporary?) to promote the creation of alternative energy sources to supplant oil. Other countries that are net oil consumers should do the same...or they should buy our alternatives.
3) Agricultural subsidies in the United States and the European Union amount to nothing less than full-scale genocide on many African countries. Predatory dumping practices combined with huge government subsidies ensure that Africans can't grow enough food to make a living, and therefore those who have land don't grow it at all.
4) This is sort of related to #1. Lack of information in non-globalized countries often fuels extremism and resentment of other countries. Tom Friedman makes the observation that no two countries that have McDonald's have ever gone to war with each other. Non-globalized countries typically know very little of the outside world, and are therefore less likely to understand the point of view of outsiders and more likely to go to war.