Expanded from an earlier answer of mine

The long and the short of it was that what happened to the fleet was it was largely sunk in the Battle of Tsushima after a long and arduous journey- seen in this map . The journey was a logistical nightmare for a Russian navy that was largely designed to fight in European waters. The inexperience of the fleet, coupled with panic over Japanese sneak attack led to various scares during the travel that the fleet was under attack. Some of this was comical, such as when the repair ship Kamchatka fell behind the fleet for engine problems off of Africa, and when she returned, mistakenly signaled she was under attack by torpedo boats. Other incidents were more serious, such as when at Dogger Bank, the fleet mistook a group of British fishing trawlers for Japanese torpedo boats and opened fire, killing three British fisherman.

The Russian fleet mostly sought shelter in French ports like Dakar, Madagascar, and Camn Ranh Bay. International maritime law allowed for belligerent warships to stay 24 hours in neutral ports for coaling, but the Franco-Russian alliance meant that the French officials often turned a blind eye to the Russians overstaying their limit, much to the chagrin of Japanese diplomats. The Russians also contracted out to private French and German civilian for colliers. In the latter case, the Hamburg-Amerika line's colliers further strained Anglo-German relations, and Chancellor von Bülow wanted the company to cancel the contract, but Kaiser Wilhelm II sided with Hamburg-Amerika as it fit into his conception of using German support for Russia as a wedge against the Franco-Russian alliance. In this instance, the French desire to keep the alliance alive and the German desire to undermine it actually worked in the Russian's favor, making the fleet's voyage possible after the Dogger Bank incident alienated an already cold Britain.

The uncertainty that Britain would honor maritime law was the reason why the main fleet did not simplfy things and transit the shorter Suez route. But Rozhestvenskii did order Rear Admiral Felkerzam's newly-christened Third Pacific Squadron to make the transit via the Canal. There were several reasons for this.

One rationale was that Felkerzam's ships were both lighter and unable to carry as much coal as the main battle line under Rozhestvenskii's flag. There was a fear that these smaller ships would hamper the already complicated logistics of the voyage. Moreover, there was also a fear that the heavily-laden battleline might also find itself aground in the narrow waters off Suez because the amount of coal taken aboard had increased their draft.

But it was more than just practical matters of logistics that underlay the division of the fleet. The same fears that inspired the Dogger Bank incident were still at play and there was a fear that the Japanese could have naval forces in the restricted waters of the Mediterranean. Felkerzam's ships were not only a tad more maneuverable, they were also expendable for the whole fleet. His ships were either older ones or auxiliaries.

The expendable nature of Third Pacific Squadron also reflected the fear that Britain could prevent the whole enterprise. Rozhestvenskii was afraid that the British control over the Canal meant they could use whatever pretext to detain or even seize the battlefleet. The 1888 Convention of Constantinople sanctified the Canal as an international waterway and guaranteed rights of access. But the Convention also had provisions governing the transit of warships and ships of belligerent powers that limited their loiter times in the Canal or coaling at Canal ports. The British, as the de facto controlling power of the Canal, had the power to interpret the Convention's provisions. They had already done so during the Spanish-American War when they narrowly interpreted the Convention to minimize the time Spain's warships spent coaling and refitting while making a transit to the Philippines. The Dogger Bank incident certainly did not endear the Russian fleet to the British and there was a fear that the British could interpret the Convention against Russia's interests. Moreover, the Convention was still a relatively untested treaty and it was unclear from Rozhestvenskii's perspective whether or not the British would abide by it or if the Egyptian government would alter the rules for coaling.

So even if Dogger Bank did not alter the Convention, the bad blood the incident created between Britain and Russia had the potential for British to use the Convention against Russia. This, coupled with the already severe logistical problems of the fleet, meant that sending Felkerzam's more expendable and lighter ships to transit the Canal was a calculated risk on the part of Rozhestvenskii and one that paid off. The British did not unduly delay or pester Felkerzam's detachment and the transit showed Britain's commitment to the Convention.

The logistical nightmare of operating a coal-powered fleet at such distances added further confusion to the fleet, as machinery broke down and men had to operate in tropical conditions they were not prepared for. The fleet's admiral forbid the opening of portholes for security against would be Japanese torpedo boats, and the result was the ships became sweatboxes. Not only men, but also officers often slept naked and the heat swelled the wooden furnishings on the ships. Fresh water was a scarce resource, so rainstorms became one of the ways for the crews to bathe and alleviate the rashes on their skin caused by the heat. Cleanliness and discipline often broke down, which was a shock to the fleet's commander Rozhestvenskii, who was a more conservative flag officer.

But despite all of these problems, the fleet made it to East Asian waters after an eight month voyage, only to sail into the Japanese battlefleet. The naturally slow pace of Rozhestvenskii's fleet made it easy for the Japanese to prepare to meet it, and Admiral Togo managed to put his fleet in a position where it could intercept the Russians. The resulting battle was quite lopsided, with the Russian ships being much less modern than the Japanese, and crewed by very tired men. The Russian fleet lost all of its eleven battleships, four of its eight cruisers, and six of its nine destroyers. In exchange, the Japanese only lost three torpedo boats.

So despite herculean efforts to get the Baltic fleet to fight the Japanese, the Russian ships were pretty much sunk by the time they arrived in Japanese waters. Now it is impossible to really know what would have happened had Rozhestvenskii evaded the Japanese fleet. Certainly Vladivostok could have accommodated the fleet over the short term; the Russian Pacific Fleet prewar was roughly the size of Rozhestvenskii's armada. But Vladivostok lacked a number of facilities that would have allowed the fleet to operate effectively despite the fact it was a relatively large harbor. This and the geographic distance from Japan or western half of Korea also meant the fleet would have likely been more of a fleet in being that would have allowed Russia to extract more concessions in the peace negotiations. Nonetheless, it would be likely a successful transit would not have been a major game-changer. The Revolution of 1905 had already broken out and reflected some of the long-simmering tensions within the Russian Empire so the government would not have been keen to keep fighting a war it had already lost with the fall of Port Arthur. Rozhestvenskii's sortie had made some strategic sense when the situation in Korea was more fluid, but the rough collapse of Russian forces in Korea eliminated whatever good Rozhestvenskii's fleet could accomplish. But the the fleet was in for the kopek and in for the ruble by the time Port Arthur fell and proceeded to its doom.