Probably the simplest way to answer this question is to take each progressive issue one at a time and describe how each of the presidential administrations addressed it.
Roosevelt took an active role in breaking up monopolies, including the giant JP Morgan. This earned him the nickname "Trust Buster." Despite this, he still advocated a free market economy. He did not oppose large businesses. In fact, he encouraged the growth of large corporations. Roosevelt felt that the federal government should only get involved when a firm used unfair practices to abuse its customers and competition. He felt that an industry on which others relied on needed to play fair. This is why he supported and signed the Elkins Anti-Rebate Act and the Hepburn Act to regulate railroad practices. He also asked Congress to allow him to use the Interstate Commerce Commission to regulate railroad rates.
Taft took an approach to monopolies that was similar to his predecessor. In some ways, he was even more aggressive. Under Taft, the federal government sued nearly twice as many trusts as it did under Roosevelt. One famous case was the Justice Department's suit of US Steel in 1910. Roosevelt issued a public criticism of this action. Overall, Taft used the courts to break up large trusts, whereas Roosevelt promoted regulation.
Wilson took an approach that combined Roosevelt's use of regulation and Tafts employment of the judicial system to target monopolies. Overall, though, Wilson felt that neither regulation nor the courts were the solutions. Instead, he wanted to promote competition among businesses through tariff reduction and banking reform. Wilson supported the Clayton Act, which defined monopolies and unfair business practices. This made it easier for the federal government to regulate them.
Roosevelt was an avid conservationist. Even before he became president, Roosevelt promoted the conservation of the country's resources and wild places. As president, he gathered together many experts on conservation. He promoted the 1902 Newlands Reclamation Act, established the Forest Service, and created numerous national parks and monuments.
Taft was also an active conservationist. He felt that public lands had a great value that should remain under the stewardship of the federal government. He supported legislation to stop public lands from being sold to private entities. Taft also created the Bureau of Mines to protect the country's mineral deposits from exploitation. However, Taft still received criticism of being anti-conservation after he fired Roosevelt's forestry chief Gifford Pinchot.
Likewise, Wilson felt that the conservation of natural resources was critical. From the outset of his presidency, he promoted protecting America's public lands. He signed the Organic Act, which further protected the national parks. However, he believed that preservation needed to take a backseat to conservation. This was illustrated when he authorized the construction of the Hetch Hetchy Dam to provide water for San Francisco.
Food and Drugs
All three of these presidents felt that food and drug safety needed to be better regulated. Roosevelt signed the landmark Pure Food and Drug Act and the Meat Inspection Act in 1906. Roosevelt felt that the federal government had the responsibility to protect American consumers from dangerous business practices and he hoped that these new laws would help do that. This led to the creation of the Food Administration. Taft also supported this and used his office to enforce the provisions of these acts. Wilson further strengthened the ability of the federal government to ensure food safety by tasking the Food Administration to vigorously enforce regulations. He even appointed Herbert Hoover as head of the Food Administration and gave him a free hand over how he conducted his affairs.
Roosevelt was well aware that there was a race problem in the United States. However, he did not feel that the government should take an active role in rectifying it. He felt that gradual change was best and that legislation was not the answer. Taft and Wilson also did little to rectify racial inequality in the country. However, Roosevelt was a champion of women's suffrage. While he did not take many actions to promote it during his presidency, his Bull-Moose Party had many women representatives on the national committee. When a referendum for women's suffrage made its way onto the New York state ballot in 1915, Roosevelt campaigned in its favor.
Taft and Wilson (at first) both opposed women's suffrage at the federal level. Taft mostly ignored the issue. At best, Wilson only gave it lipservice in his early years as president. However, the contribution of women during WWI convinced him that they could play a bigger role in the public sphere. By 1919, Wilson became a more vocal supporter of the suffragette's cause.