U.S. answer only.
Are there a few database-like resources that would allow me to easily
search for such cases?
It is balkanized. It also isn't entirely clear what information about the cases interests you.
Almost all federal court filings (but not administrative law decisions) are in a database called PACER.
Each state has its own system. Some are almost completely unified, and in others, there are many databases.
In Colorado, for example, the Colorado E-Filing system has all filings in state courts, but courts outside the state court system, mostly the Denver County Court and municipal courts, as well as some major private arbitration firms, are on their own and most subcontract the job to a division of LexisNexis, a private firm. In both cases, access to these filings is not free except to parties, and has lots of data with access restricted to parties and the court. Published appellate court decisions are also available at an Internet based source. None of these covers administrative law decisions in Colorado, however.
Administrative law decisions are usually kept by the agency and also often by a commercial firm that compiles them. Some of them do not give the public access to the decisions in the absence of a FOIA or open records act request.
Commercial firms like Westlaw and LexisNexis and several less well known firms (including a free one run by Cornell University), keep databases of published decisions of appellate courts plus a somewhat random assortment of unpublished decisions, with federal court trial court decisions getting much more heavy coverage than state court trial decisions and unpublished state appellate court decisions. But, these are only key court orders, not all filings in the case, and are not the true source documents.
There is also a non-profit consortium that maintains a database of court records from the 75 most populous counties in the United States, and there is a private firm that keeps selective track of jury verdicts to the fullest extent that it can obtain them.
Some credit reporting agencies (both consumer credit and business credit agencies) maintain databases of judgments and liens.