This answer from /u/iwinagin is narrowed to only a British reaction but may help give some insight:
Edit: Fixing Username.
This is one of the first times I’ve seen a question that I might be able to contribute to, so I’ll give it a shot!
In brief, the answer to your question of what the United States got out of sending the Great White Fleet around the globe was a little bit of everything.
Regarding whether this was a technical/logistical achievement, Roosevelt noted, in a letter to Lawrence F. Abbott on September 13, 1907, that “if this enterprise is carried out it will represent a far longer cruise than has ever been made in modern times by a battleship fleet of even half the strength.” https://www.theodorerooseveltcenter.org/Research/Digital-Library/Record?libID=o200296 So that certainly seems to have been a component of this.
Similarly, in a letter between Admiral Robley D. Evans and Willard H. Brownson of August 17, 1907, Robley notes that “of the present fleet of 16 battleships, ten have never performed a maneuver together,” and emphasizes the importance of doing preliminary drills. https://www.loc.gov/resource/mss38299.mss38299-076_0001_0857/?sp=334 This seems to suggest that Evans knew that this had the potential to be a difficult journey, showing again that its completion would be seen as a feat of organization and seamanship.
By early 1909, Roosevelt comments, in a letter to German Emperor William II, that “in gunnery and in battle tactics no less than in the ordinary voyage maneuvers, there has been a steady gain; and the fleet is far more efficient, collectively and individually, now than when it left these waters over a year ago.” https://www.loc.gov/resource/mss38299.mss38299-353_0130_0654/?sp=455 So this certainly seems to have been a result of the voyage, if nothing else.
Domestically, the movement of the fleet was used as a matter for celebration, with Oregon Senator Jonathan Bourne transmitting a telegraph he received from the Portland Chamber of Commerce to President Roosevelt advertising the city as a site for the fleet to visit, and referencing the visit of the fleet to Oregon as a favor - https://www.loc.gov/resource/mss38299.mss38299-076_0001_0857/?sp=8
In a somewhat similar vein, Roosevelt writes to Edward Robeson Taylor in February of 1908 about the Great White Fleet’s arrival in San Francisco, and connects the arrival of the Great White Fleet with a celebration of (and in) the city - https://www.theodorerooseveltcenter.org/Research/Digital-Library/Record?libID=o201813
Internationally, Roosevelt certainly viewed the fleet as a tool of international relations. On July 30, 1907, Roosevelt wrote to Ambassador Henry White that “the Japanese yellow press is showing itself to be quite so obnoxious as your yellow press at its worst, and I think it is high time for our fleet to visit the Pacific. I am exceedingly anxious to impress upon the Japanese that I have nothing but the friendliest possible intentions toward them, but I am none the less anxious that they should realize that I am not afraid of them and that the United States will no more submit to bullying than it will bully.” - https://www.loc.gov/resource/mss38299.mss38299-346_0300_0827/?sp=198
Similarly, in a letter of November 19, 1907 from Roosevelt to Ambassador Charlemagne Tower, he writes that “I can hardly believe that Japan is intending to strike us, but I am taking and have taken every step to be ready,” and continues on, writing about the movement of the Great White Fleet to the Pacific - https://www.theodorerooseveltcenter.org/Research/Digital-Library/Record?libID=o200817
The movement of the fleet certainly seems to have been viewed with some trepidation by Japan at first, John Callan O’Laughlin suggesting, in a letter to Theodore Roosevelt of November 4, 1907, that a recent action by the Japanese could be interpreted as trying to “restore Japanese prestige, which had been affected by the dispatch of the fleet to the Pacific.” - https://www.theodorerooseveltcenter.org/Research/Digital-Library/Record?libID=o57370
Roosevelt also certainly felt some hesitation about sending the fleet to Japan as well, as he cautions Rear Admiral Charles S. Sperry about avoiding any “suspicion of insolence or rudeness” while the fleet is in East Asian waters in this letter of March 21, 1908 - https://www.theodorerooseveltcenter.org/Research/Digital-Library/Record?libID=o202062
Apart from these relations with Japan, and the enthusiastic response of Australia to the visit of the Great White Fleet, which was mentioned in another post on here, the fleet seems to have been praised by several countries (at least in the documents I have seen), as with this letter from German Emperor William II to President Roosevelt praising the punctual arrival of the fleet https://www.theodorerooseveltcenter.org/Research/Digital-Library/Record?libID=o299324 or with this toast by Brazilian President Afonso Pena, who offers a toast to the “new and magnificent demonstration of the unsurpassable vigor and of the extraordinary energy” exhibited by the United States, which he claims as a fast friend of Brazil - https://www.theodorerooseveltcenter.org/Research/Digital-Library/Record/ImageViewer?libID=o299213&imageNo=2
(This is somewhat balanced, however, by rumors of a plot to sabotage the ships as they travel along the South American coast, as mentioned by Ambassador Henry White to President Roosevelt in a letter of January 3, 1908 - https://www.theodorerooseveltcenter.org/Research/Digital-Library/Record?libID=o57909 - so it cannot be said that the fleet was uniformly loved)