This is going to be a region-specific question, and as I'm most familiar with the Caucasus, I'll answer for there, and note that what happened there may not have been the same for other regions.

The February Revolution, which saw the end of tsarism in Russia, with Nicholas II abdicating on March 2 (under the Old Style dates, which was used throughout the Russian Empire and subsequent territories until 1918; this was 13 days behind the New Style used in most of the rest of the world). The next day a telegram was sent to Noe Jordania, the leading Georgian Menshevik in Tiflis (now Tbilisi, the capital of Georgia), the seat of the Caucasian Viceroyalty and main city of the Caucasus. It said, in Georgian, "მთავრობაძე გარდაიცვალა. შეატყობინეთ ნათესავებსა და ახლობლებს" (Government-son is dead. Notify friends and relatives). Government-son was a code for the end of the tsarist government, and meant that the Viceroy (Nicholas II's cousin, Grand Duke Nicholas) was effectively powerless. With that in mind Jordania quickly moved to establish Menshevik (a socialist group, not quite as reactionary as the Bolsheviks) control in the region.

A week later, on March 22, the newly-formed Provisional Government in Petrograd (the Russian capital) set up a replacement for the Viceroy, a Special Transcaucasian Committee (known by the Russian abbreviation, Ozakom). It was sort of a collective viceroyalty, as it had several members on it, a mix of members of the Duma (Russian parliament) and some other local political leaders. Much like in Petrograd, which had a "dual power" system (the Provisional Government "worked" in tandem with the Petrograd Soviet, meaning it didn't really work at all), the Caucasus had a dual power structure, with the Tiflis Soviet having popular support, being a representative body of elected members (like in Petrograd). The Ozakom also lacked any serious authority as it had little support from Petrograd, which was busy dealing with everything else that was going on at the time; the Provisional Government simply didn't have time to worry about a provincial backwater, so left things to their own. That said, the Ozakom and Tiflis Soviet worked to remain part of Russia, albeit with some stronger autonomy; independence was not in the plans for anyone there.

This system existed until October 1917, when the Bolsheviks staged the October Revolution, and took over Petrograd and broadly Russia as a whole. Wildly unpopular in the Caucasus, the Bolsheviks were seen as usurpers and their authority not recognized. Concerned that the Bolsheviks would try and use the Ozakom to direct things in the Caucasus, a meeting of representatives from there, the Tiflis Soviet, and other relevant bodies decided to simply end the Ozakom and replace it with something new: the Transcaucasian Commissariat, which would thus have no legal reason to follow orders from Petrograd. However this was still just meant to be a temporary measure, as there was a planned Constituent Assembly planned for Petrograd in January 1918, and it was hoped that the Bolsheviks would be forced out at that, and order restored. Autonomy was thus still the goal, and independence still not desired.

On the afternoon of January 5, 1918 the Constituent Assembly had its first, and only, meeting, lasting until the early morning before being broken up by the Bolsheviks. This confirmed for most people that the Bolsheviks were here to stay, and plans were adjusted. In the Caucasus, they were also facing an imminent invasion by the Ottoman Empire, and with the Bolsheviks simultaneously starting negotiations at Brest-Litovsk to end the war, began to work to free themselves. A parliament, the Seim, was formed, and efforts to negotiate a separate peace with the Ottoman were launched, but stalled as the Caucasus was still de jure part of Russia.

The Treaty of Brest-Litovsk was signed on March 2, and effectively gave up large parts of the Caucasus to the Ottoman. As no one from the region was invited to the treaty talks, they were understandably not happy with this decision, and refused to acknowledge it. The Ottoman didn't care, and simply launched an invasion. Peace talks were started in Trabzon, in eastern Anatolia, but the Ottoman again refused to properly negotiate as they weren't talking to a legitimate state. Faced with this, the Seim declared independence on April 22 as the Transcaucasian Democratic Federative Republic (TDFR), which quickly worked to negotiate a ceasefire. For reasons beyond this question that didn't really work, and it took German intervention (via stationing troops in Georgia) for that to work.

An unstable coalition of Armenians, Azerbaijanis, and Georgians, the TDFR was never something that would be long-lasting. That the three groups had differing goals (Armenians were adamantly opposed to the Ottoman, given the recent genocide and swarms of refugees from said genocide; Azerbaijanis, as fellow Turks, supported the Ottoman; Georgians were simply trying to not be conquered by a new imperial power) it broke up after five weeks, when the Georgians declared independence on May 26 (Armenia and Azerbaijan did the same 2 days later). All three states sort-of survived for the next couple years, before being invaded and annexed by the Red Army in 1920 (Azerbaijan and Armenia) and 1921 (Georgia).

You'll see I didn't bring up much about the Provisional Government (what you refer to as the "Russian Republic of 1917"), simply because there was not a lot they did here. However throughout the few months it was around, the goal never was for regions to break free, and for them to stay part of Russia. The empire was not supposed to break apart, just simply change government from tsarism to something more democratic. But owing to more pressing issues, like the ongoing German invasion, an abortive Bolshevik uprising in July, and general instability, the Provisional Government was never able to really do anything with the Caucasus. It would be the same story in most other regions: Poland and Ukraine were largely occupied by the Germans and Austrians by this point (a puppet Polish state was even established in 1916, though it never amounted to anything); Finland had already been an autonomous region that was de facto independent prior to 1917, so had little impact on its status; Belarus was not really a thing yet (the Germans would actually help set up the first independent Belarusian state in 1918); the Baltic states (Estonia, Latvia, Lithuania) were either partially occupied by the Germans, or run by ethnic German nobles; and Central Asia, known at the time as Turkestan, was dealing with its own revolts (both Bolsheviks and Basmachi, a Muslim group opposed to the conscription introduced into the region in 1916).

In short, there really wasn't much that was done, and it's hard to say where things would lead to. It's quite possible that further autonomy would have been given, as that was the main goal of nearly all non-Russian groups, but with so much going on it is hard to say how that would have worked out in the end, if it was ever really possible.