Which area of the world used the most effective method to challenge and change European domination in the 1800s?
I will assume that we are treating countries like the United States and Australia as European countries for the purpose of this question.
If so, there was no one area in which all countries were effective in challenging and changing European domination. Perhaps the most effective country at this was Japan. Japan was forcibly "opened" in 1853 but soon went about modernizing and Westernizing. This was the most effective way to challenge European domination. Japan essentially became like the West in many important ways, allowing it to become a power that was in the same league as almost any country in the world.
Japan, with the era Meiji in the 1860ies, began to modernise the country : industry, army, fleet.
So Japan could invade Taiwan (Formosa), Korea at the end of the century, than fight and sink the Russian fleet in 1904.
Japan had decided to fight white countries on their own field, and succeeded, until it was TOO MUCH in the years 1930 and 40ies for the others, specially the USA.