Here are a few answers I've given in the past that have discussed some of what happened to the Jews who were expelled from Spain or left Spain. If you want any follow-up info, hit me up! Also, a great starter book on the subject is The Jews of Spain by Jane Gerber, which goes through the entire history of the Jews in Spain and after their official expulsion.

First, it's worth noting that a large share of Spanish Jews chose to convert to Christianity and stayed (sometimes sincerely, sometimes practicing Judaism in secret). Many of them ended up leaving much later and returning to Jewish communities, even after centuries of being Jewish only in hiding. But some simply blended in over time into Spanish and Portugese populations--a large chunk of Hispanic people likely have at least one Jewish ancestor, and some are descended from people who maintained Jewish practice for hundreds of years, which eventually may've eroded into a few odd family customs. Some even left and returned to Spain a few years later. Another note is that a huge proportion of Spanish Jews fled to Portugal but were expelled from there a few years later, and ended up in largely the same situation--but sometimes there are references to the "Spanish and Portuguese community" for this reason.

A note on terminology, Jews from Spain are often called "Sefardim", though this term is used to broadly encompass Spanish and all non-European Jews--"Western Sefaradim" is more specific, but may not include North African Jews who can be a mixture (see below). Another common term is "gerushei sefarad" (expellees of Spain).

Anyway, to actually answer your question: there were several major destinations. The primary one was North Africa. Still under Muslim control it was generally a friendly place for Jews to go, but being refugees they were still vulnerable to and suffered abuse. There was already a large Jewish population in North Africa, and as there had been outbreaks of anti-Jewish violence in Spain before 1492, there had been Spanish Jews going to North Africa for a long time. Most famous of these figures is Maimonides, who was a prominent Rabbi and philosopher (and doctor), who grew up in Spain but fled from persecution by the Almohads (an anti-Jewish Muslim group in North Africa and Spain), settling in Egypt. The bulk of North African migration didn't go that far east, the North African population of Spanish Jews mostly was in Morocco, Tunisia, and Algeria.

Some also moved east but stayed on the north shore of the Mediterranean. There were Spanish-origin communities in Italy, where there also were existing Jewish communities. Many more moved into areas under Ottoman control, which was friendlier to Jews. The Balkans had a large Sefardic population, as did Greece, as did Turkey. A few "closed the loop" as it were and moved to Israel--most famous of these is R' Yosef Karo, great codifier of Jewish law, who was expelled with his family as a small child and migrated through the Balkans, eventually ending up in the Galilee.

Others moved more north. Some moved to England. The exact origins of this community are a big murky, and it's likely most of them practiced Judaism in Spain in secret, immigrating to England when Jews were re-admitted in the 17th century (a large share of this population was Portuguese). Many of them ended up in the Netherlands, which was a Spanish possession. My knowledge here gets a bit murky, but there is definitely a Dutch-Spanish Jewish population. I've heard (which means, take with a grain of salt) that some moved to Eastern Europe and integrated with the population there. It's believeable, since there's geographic proximity between the northeastern extent of the Balkans and the southwestern end of Ashkenaz-land, but the sources I have on the Spanish Expulsion don't mention it.

From the above, some moved to the Americas. Some founded communities in the Caribbean or in Brazil. And there were (are) communities of Western Sefaradim in America, mostly via the Netherlands and/or Britain. Some of the US's oldest synagogues were founded by Sefaradim. Though their numbers were fairly quickly overwhelmed by Central & Eastern European Jews, some synagogues retained the Spanish/Portuguese liturgy, and still do today. And as the longest-established Jews in America, many of American Jewry's prominent members were Spanish until the Eastern European community was well-established.

For a bit of an intermission, one way this is reflected in Jewish history is that large numbers of Jews used Judeo-Spanish (various Jewish "versions" of various Iberian languages, often called Ladino, but there's a whole bunch of non-identical varieties with names I don't completely know, so I'm gonna stick with Judeo-Spanish). The language(s) are similar to Spanish, but have diverged over time, usually were written in a variant of the Hebrew alphabet, and borrowed words from the local language after the expulsion. There were huge populations of Judeo-Spanish-speaking Jews in the Balkans, Turkey, and North Africa. A major compelation of Midrashim was composed in Judeo-Spanish, the book Me'am Lo'ez.

As for their descendants, a large number are still alive today. There are Spanish communities in a few cities in America and in London, and synagogues that follow the Spanish liturgy. But most Sefardic Jews ended up in the other places listed above and have since migrated to Israel. There's been a lot of homogenization among the non-Ashkenazi Jewish population in Israel, but there are still people who are identifiably of Spanish-Jewish ancestry, and from communities where there's a significant proportion of Spanish Jews but individuals aren't identifiable as such necessarily. There's the caveat that there are still Jewish communities that are identifiably Spanish in Turkey and perhaps Greece, and I believe Italy as well--so while most of the Sefardic Jews of the Mediterranean are in Israel which can be a Jewish melting pot, that's not true universally.

Sources and further reading:

Chouraqui, André, and Michael M. Bernet. Between East and West: A History of the Jews of North Africa. New York: Atheneum, 1973

Chazan, Robert. Reassessing Jewish Life in Medieval Europe. Cambridge: Cambridge University Press, 2010.

Mitchell, Bruce. “LANGUAGE USAGE IN ANGLO-SEPHARDI JEWRY: An Historical Overview of Spanish, Portuguese and Judeo-Spanish in England from the Expulsion to the Present Day.” European Judaism: A Journal for the New Europe, vol. 33, no. 1, 2000, pp. 99–108.

Elon, Menachem. “THE CONTRIBUTION OF SPANISH JEWRY TO THE WORLD OF JEWISH LAW.” Jewish Political Studies Review, vol. 5, no. 3/4, 1993, pp. 35–54.

Lamdan, Ruth. “Jewish Women as Providers in the Generations Following the Expulsion from Spain.” Nashim: A Journal of Jewish Women's Studies & Gender Issues, no. 13, 2007, pp. 49–67.

Benabu, Isaac. “WESTERN JUDEO-SPANISH (HAKITÍA): TRACING SPEECH THROUGH NARRATIVE.” European Judaism: A Journal for the New Europe, vol. 44, no. 1, 2011, pp. 36–50.

Kamen, Henry. “The Mediterranean and the Expulsion of Spanish Jews in 1492.” Past & Present, no. 119, 1988, pp. 30–55

Ray, Jonathan. “Iberian Jewry between West and East: Jewish Settlement in the Sixteenth-Century Mediterranean.” Mediterranean Studies, vol. 18, 2009, pp. 44–65.

Here are some other links:

A guy speaking Judezmo, from Thessaloniki

One of the more famous Judeo-Spanish songs, Cuando el rey Nimrod

A Judeo-Spanish religious song for after meals. Note the choirmaster wearing the top hat--some British synagogues, especially S&P ones, and some S&P synagogues elsewhere, have maintained the custom of synagogue functionaries wearing top hats. Here's a video with (even more of that](https://youtu.be/oHvH1T5Q_ds). This exactly tune is a British S&P one, which has made its way around American Jewish communities without any particular S&P connection, because of the role of S&P Jews in the founding of early American communities

One of my personal favorite bits of liturgical music, from the Anglo-S&P community