Because your other questions have been about the early 1900s (or earlier) I am assuming that you are asking about the first President Roosevelt. I have edited and moved your question to reflect this.
Theodore Roosevelt had a number of major foreign policy initiatives and actions during his time as president. Some might argue that not all of these should be called achievements. These included:
- Establishing the United States as a relatively important world power. This can be seen in at least two ways. One was the Roosevelt Corollary to the Monroe Doctrine. Here, Roosevelt asserted the right of the US to intervene in the domestic affairs of any nation in the Americas to prevent European interference. The second was the sending of the "Great White Fleet" to show American power around the world.
- The beginning of the Panama Canal. Under Roosevelt, the US helped (in somewhat dubious circumstances) Panama become independent of Colombia. The US then negotiated a treaty for a canal zone and started work on that canal.
- The Russo-Japanese War. President Roosevelt brokered a peace treaty between Russia and Japan in 1905. He won the Nobel Peace Prize for his efforts.
These are the most important events (that could be called achievements) in Roosevelt's presidency.