Let's say I wanted to generate a report on the trends in outcomes of
civil cases. Is there anywhere I could download any kind of data
related to it? The parties involved, maybe information about the size
of compensation, result of the case etc.
The raw data is a matter of public record, but it isn't available in a downloadable format. You have to go court by court, or case by case, look at the public records, code and classify it, and then analyze the data yourself.
Some summaries are available in annual reports of the federal court system and of many state court systems, but this typically looks at total dockets, and dockets by type of cases, but rarely looks at outcomes. The most detailed outcomes you are likely to get from these sources are number of cases terminated in a year (often by geographic subcategory), number of jury or bench trials with some division by type of case, number of defendants adjudicated, and number of dismissals prior to trial.
The U.S. Sentencing Commission does more analysis of outcomes from detailed case data.
Many professors writing articles for academic journals do it the hard way, and there are some collaborations of professors and research institutes that collect large, detailed (although not complete) databases, for example, limited to federal courts (where it is easy to get data on individual cases in PACER) or to, for example, 75 most populous counties with cooperative custodians of records. But, often you need to be an accredited member of the collaboration working on that to get access to the raw data.
One commercial source is a firm called the Jury Verdict Reporter that regularly publishes detailed data on as many jury verdicts as it can, but this is quite an expensive resource to use.