A useful starting point for searching for data is secondary literature.
By reading sources you gain insight in the topic. This helps you to come up with good keywords to search with. It also shows you the authors who are considered specialists on your chosen subject.
Scholarly literature may contain visualizations of data such as graphs, tables, and references to data in printed or in digital form. Sometimes a link to the dataset is provided, giving access to the dataset but also showing in which data archive it is published. This data archive can contain more relevant datasets for your research.
Besides using references to data, you can also search if the data collected for the study you are reading is published in printed or digital form (as in the example below).
To sum up, secondary sources about your topic are useful to search for relevant datasets in multiple ways:
In the same search engines where you find citations of secondary sources, you can search for the sources using datasets. You can use keywords related to your topic combined with the keywords: datasets OR "data set" (both forms are common in English). You can try again with more detailed keywords related to data: like "statistics OR empirical OR quantitative OR numeric" or "graph OR table OR figure" etc...
An article can contain useful data visualizations, as tables and figures. Sometimes the underlying dataset used for the publication is published online.
Take for example the article From Baghdad to London: Unraveling Urban Development in Europe, the Middle East and North-Africa, 800-1800. In this article tables and figures are found, but the underlying dataset is also published online. To find this dataset a few steps need to be taken, which are given below.
Step 1. In this article, you will find different visualizations of data: tables and figures. This is already relevant data, but the text does not contain a link to the underlying dataset.
Step 2. The article refers to a supplemental appendix where the collected data is documented in detail. The appendix contains no link to the dataset, nor a citation of the dataset. You will need to set an extra step to obtain it.
Step 3. Plain Google search: try and combine the title of the article with the keyword "dataset". This is a direct search, as you a searching for a dataset of a know article. You will directly find the link to the original data used by the authors of the article.
Step 4. The dataset you find via Google is archived on the site of the Harvard Dataverse data repository. The title is: "Replication data for: From Baghdad to London: Unraveling Urban Development in Europe, the Middle East, and North Africa, 800-1800". On top of the page, you can find the title of the dataset and how you can cite it. In three tabs, you can find the datasets and additional information.