What was life like in the Ottoman Empire for Christians and Jews?

This varied HUGELY. The Ottoman Empire existed from 1299-1922. For most of the Empire's history, until the reforms of the 19th century which eventually guaranteed equal citizenship, a lot of governance was conducted through "millets" ("nations" is one possible translation, but "religious group" or "ethno-religious group" is probably the best translation). The four basic millets for most of the Empire were Muslim (including Turks, Kurds, Arabs, Albanians , etc), Greek (including all Orthodox Christians, including Arabic and Slavic speakers), Armenians (Armenian Apostolic--by the end of the Empire, once European missionaries became active, there were also separate Armenian Catholic and Protestant millets), and the Jews. However, as the Empire grew so did the number of millets. Karen Barkey has argued that the main strength that Ottoman rule had was adaptability. The exact operations of the millets was complex though. For instance, some aspects were formalized for some millets in the 15th centuries, and wouldn't formalized for others until the 18th or even 19th centuries, but still had been informally in place since at least the 15th centuries.

But, for the most part, the Ottoman system of rule meant that people could live unmolested within their own communities. You walk around neighborhoods of Istanbul and you can still see signs of once thriving Armenian, Greek, and Jewish communities that have largely disappeared (there are something like 100,000 Armenians in the city still, but maybe 10,000 Jews and fewer Greeks). There's still a famous Jewish hospital near the old Jewish neighborhood of Balat, for example, and a famous abandoned Greek orphanage on the (historically Greek) Islands. A very prominent hospital almost adjacent to Taksim square has a very Armenian name. For the most part, the millets were able (and chose to) build their own institutions--schools, hospitals, etc. There are still legally separate Jewish, Greek, and Armenian schools, for instance. Some Istanbul neighborhoods (Galata, for instance), at least by the 19th century, were quite a bit more mixed, though others were long historically associated with one minority or another (Fener, the Greeks, Balat, the Jews, Galata was historically associated with the Levantine Italian Catholics, who also still exist in the city in tiny numbers--one of my landlords was a Levantine Italian who said his family had been in the city for 500 years at least). Outside of the major cities, you see the same millet system, but as peasants had few recourses to formal institutions (even villages mosques seem rare before the 19th century; there seem to be more village churches, but I am not sure--Jews were obviously mostly urban). One of my friends is writing his dissertation on Eastern Anatolia (cites like Muş and Elazığ), and is arguing that the meaningful divisions, in terms of local political power, even at the beginning of the 19th century were not Christian (mainly Armenian) and Muslim (mainly Kurdish and Turkish), but mountain and valley, and he sees a lot of records of Kurdish mountain tribes and Armenian mountain tribes cooperating against valley Kurdish and Armenian groups. Only in the 19th century, as the Ottoman state exerts a higher degree of centralized rule, do we see the salient division of local political power rebracket along ethno-religious (and later simply ethnic) lines. This fits well with Michael Hechter's argument in Containing Nationalism that nationalism's pretty sudden emergence globally in the 19th century is tied to state's shift from indirect to direct rule.

In general, we can pretty high level of confidence say that life for Jews in the Ottoman Empire was probably better than life in Christian Europe, until probably at least European emancipation. Blood libels, for instance, which spread periodically through Europe did not come into the Ottoman Empire until the early 19th century. While there increased through out the 19th century, the Empire generally had less anti-Semitism than Europe (and much of it, interestingly, was Christian on Jew rather than Muslim on Jew).

For Christians, they were generally allowed to live separate lives, but there were, of course, downsides. They experienced in the Ottoman Empire legally inequality (Jews obviously experienced this everywhere until Emancipation) and Muslim legal codes disadvantaged non-Muslims in matters of inheritance, etc. such that occasionally you see one member of a family convert and this sets of a chain reaction of conversion so that the patrimony does not end up going to that one person. This, though, was not excessively common and we have plenty of records of Christians and Jews voluntarily going to Muslim courts to get a better a ruling than they would have gotten in their own millet's courts. The from the 14th to the 17th centuries, devşirme system of kidnapping Christian boys from the Balkans for military purposes was an obvious downside to being a Christian, such centralized extremely exploitative measures based on millet are relatively rare in the Ottoman Empire, which generally thrived on diversity (Karen Barkey calls it an "Empire of Difference"). As nationalism increased within the Empire, starting especially with the successful Greek independence in 1821, so did tensions. This century of nationalism culminated with the systematic murder and expulsion of the Christians of Eastern Anatolia (not just Armenians, but Greeks and Syriacs as well) during World War I, at the very of the Ottoman Empire. Eventually, as the Empire was transitioning to the Turkish Republic, even most of the Christians of Western Anatolia were expelled (this, it should be noted, should be put in the context of almost a century of Muslim flight and expulsion from the Caucasus as the Russian state moved south, and at least fifty years of Muslim flight and expulsion of Muslims from the Balkans as Christian-majority nation states broke away).

These 19th and 20th century manifestations of nationalist movements and their antagonists shouldn't be read back further, however. At the height of the Empire, there would be, for example, generally Greeks and Jews at court (the court physician, for example, was generally Jewish, and the Dragomans who conducted much of the Empire's diplomatic affairs were generally Greek). The treatment of Christian peasants varied with local rulers, though generally I would say that their exploitation wasn't particularly based on religion for most of the Empire at most points in time. Even into the 20th century, most important private businesses (our records suggest not just many, but most) in the Empire were owned and operated by Christians and Jews, especially as international reoriented in the 17th and 18th centuries (as the Age of Sail made traditional overland routes less prosperous) and as various Christian groups and eventually the Jews fall under the protection of Europe powers.