Gauging the 'domestic reaction', or 'public reaction' to any given thing is no simple feat. Because this involves the feelings and beliefs of millions of people in most cases. In Turkey of the time, there's the additional problem that most people in question were illiterate. If we are talking about villagers in Anatolia, for instance, good luck with trying to see what they thought of the new regime through the diaries which they never wrote. But we do have some insight into overall trends. And this is what I will try to provide.
In large towns and cities, when considering educated people, especially young and educated people, the reaction seems to have been mostly positive. For many women in these places, the reform meant emancipation. They could go out without necessarily wearing a burqa or anything of the sort. The police were regularly instructed to prevent men from verbally (or, of course, physically) assaulting women on the streets. They could venture into professions that were impossible for them before. If lucky enough, they could become a member of the parliament, judge, or indeed, a war pilot! Things were probably a bit different in smaller towns and villages, especially the further east you went in the country. Though there was often no serious threat to the new regime in these places, nor was it able to thoroughly penetrate the masses there. The result was often a precarious balance where the state refrained from mingling with their affairs as consistently as in cities, and the locals accepted the state authority as expected.
Besides this, it does appear that some central tenets of the new regime and especially the leader himself, Mustafa Kemal Atatürk, did enjoy widespread popularity. This is hardly surprising for a number of reasons. Probably the most prominent one is that Atatürk was the leader of the Turkish War of Independence (1919-23). That seems to have given him a lot of credit in the eyes of many Turks.
But not everything was rosy. In fact, we know more about discontent. This is not necessarily because it was more common, but because the state and basically anyone else had their interest piqued more by trouble, and it is through their records that we learn the past. By way of a single example, the Italians were understandably more interested in rumours or facts about discontent in a region of Turkey than repeatedly receiving reports on how everything seemed OK. Who cares about the latter, right?
The case of Kurdish revolts is probably the most well-known case of discontent. From the War of Independence to 1938, there were intermittent revolts and clashes in the eastern and southeastern sections of Turkey (1921 Koçgiri, 1925 Sheikh Said, 1930 Ağrı, 1937 Dersim among others). Now, revolts did occur before the Republic, and before the Young Turks; so it is wrong to think the entire problem (some) Kurds had was with reform. Some simply did not want to pay taxes or conscript. However, it does appear that reaction to new reforms, especially to the ones having to do with religion and secularism, partly motivated Kurdish revolts under the Republic. With the lens of today, many scholars see the revolts as nationalist attempts for independence, and it is likely that these concerns played a role. But, if you look at the Sheikh Said rebellion of 1925, for instance, it is clear that religion and restoration of the Caliph were quite important for the rebels. So, here's one section of the population that did seem unhappy with the new regime's reforms: portions of the Kurds in the rural areas of the east. For more information on the role of religion in these revolts, the role of religion in Kurdish culture, as well as how this evolved and changed over time, I'll refer you to the work of Martin van Bruinessen. He concludes that especially in the Sheikh Said rebellion, the role of religion was no less important than nationalism.
We have reason to think that there was some discontent with the secularist policies in other sections of the country too. A particularly pointed example is provided by the notorious Menemen incident from December 1930. A small group associated with a branch of the Naqshbandi order took to the streets Menemen, a town near İzmir. This is one of the westernmost places in Turkey. Mind you, İzmir has been deemed one of the most open/cosmopolitan places in entire Turkey ever since the mid-1850s. It is said that these individuals 'waved the green banners of the Caliphate' and called the population of Menemen to join their revolt. A young officer called Mustafa Fehmi Kubilay stumbled upon these and ordered them to dissipate. They attacked Kubilay and his men. Until the reinforcements with serious arms arrived from the barracks, the rebels had taken hold of Kubilay and killed him. The reports suggest that Kubilay was then beheaded, with his head placed on a spike and carried around the town until reinforcements came and opened fire at the rebels. There's much controversy as to the true nature of the event, and some even question whether this was really an Islamist rebellion of sorts or the work of an insane man who thought of himself as the Messiah. However that might be, the news that a Turkish officer could be beheaded by pro-Caliphate folks near İzmir created an outrage. Some memoirs suggest Atatürk even intended to declare Menemen a ville maudite, infuriated at not only the event but also the perceived passivity (if not support) of the city's inhabitants towards the rebels.
Months before the Menemen incident, there were again reported 'green banner' sightings in İzmir and Balikesir, another western province just to the north of İzmir. These had to do with the emergence of a liberal-ish party of opposition, the Free Republican Party (the FRP). The founders of the FRP had nothing to do with Islamism: the founder Fethi Okyar, who actually founded the party upon Atatürk's instructions, was an old friend of Atatürk and well-known for his secularism. But the Republican People's Party (the RPP) circles long suspected a growing, bottom-up Islamist infiltration of the party. The RPP archives are full of reports and inquiries between provinces and Ankara on suspected or actual infiltrators. With Okyar on an election tour in the Aegean region, there came the reports of 'green banners', as well as 'propaganda' (again, perceived by the RPP circles to be bottom-up and stirred by Islamists) that the FRP would restore the Caliphate and the old script. The tour ended with large-scale clashes and incidents, and Atatürk asked Okyar to close the FRP soon after. To what extent did these reactions have to do with a reaction to the regime and reform? I think it is fairly clear that there is some truth in all that. The idea that people were unhappy with the corruption of local party elites and therefore thought becoming an Islamist would be jolly good seems odd. But we have to also consider the global context: this was when the Great Depression hit Turkey, with İzmir and Balıkesir, with their agricultural economies especially linked to world markets, being hit particularly hard. So perhaps most people turned to the FRP simply because of local corruption and economic troubles, whereas some others did in fact wanted a return to the old times. We know that, when the government decreed the call to prayer had to be performed only in Turkish in 1932, a non-negligible number of people tried to ignore or contrast this and kept the Arabic call to prayer. The archives are again full of details on how the police and the gendarmerie found these people and took them to the court, with cases spread all across the 1930s, including times when the economic situation did get better. Likewise, there were occasional local attempts to open Quran instruction courses where children would be taught the Arabic alphabet to read Quran. These, inter alia, seem to reflect discontent with especially the cultural-religious policies of the Republican regime rather than unease with corruption or the economic situation.
So, risking overgeneralisation, we can conclude that the more urban, educated people tended to embrace the new regime. The rural people were not as 'reactionary' as feared by some RPP circles, and their 'rebellions' sometimes had more mundane causes. But, especially in certain rural areas indeed, there was a reaction to the new policies of the Republican regime, especially regarding religion and secularism.