Google, Google Scholar, and Wikipedia (follow up on the citations supporting the articles) are often decent tools.
Frequently, they will also send you to the website of a federal agency which has links to relevant legal resources, to law review articles on topic, and to independent third-party discussions often from law firms or non-profits that deal with legal issues in the industry. Sometimes medical or health industry periodicals will discuss legal topics as well. Another great resource when you can find it are Congressional Research Service reports (this is an independent research arm of Congress related to the Library of Congress) and publications of the relevant Congressional committees at their respective websites.
A law review article on a topic will typically comprehensively review all relevant legal authority on the subject matter of the article through the data of publication (minus two or three months) all in one place with explanatory commentary and is a gold mine if you can find it, although older ones often get outdated in fast changing areas of law like health care law. Law review articles often put in context a newly decided appellate court or U.S. Supreme Court case, a newly enacted statute or amendment to it, a major overhaul of regulations or new regulations where there were none before, or a policy analysis over the overall legal framework of an area of law.
Some large complex federal health care programs like Medicare, Medicaid, and the Affordable Care Act also have a level of agency guidance below regulations containing agency determinations that are not as general in application as a regulation, but usually those are only available in a digestible manner with an expensive specialty law subscription. Also, note that most operative Medicaid regulations are at the state rather than the federal level in this joint state-federal program with the federal regulations mostly stating what states can and cannot put in their state regulations.
The biggest thing that the resources you mention are missing from statutes and regulations is case law. If you have the name of the case, a date or year of a decision, and any additional citation information, a variety of sources (e.g. Findlaw) will find the text of the case for you. But, almost all of the leading word search based ways to locate statutes and annotated statutes and regulations that list cases that have interpreted a code or regulation section are on a paid basis.
Case law is basically indispensable in researching U.S. federal health care law. For example, the meaning of several key provisions of the Affordable Care Act have been the subject to extensive litigation, in some cases all of the way to the U.S. Supreme Court or with conflicting resolutions from different appellate courts. It is impossible to discern the statute's true legal meaning without the interpretive gloss of case law.
Your university library may have a subscription, however, or might be able to get you temporary access to an affiliated or reciprocity cooperation based law school library with student legal search database access on a temporary basis.
Short of an online research database that is usually paid, your best shot is probably to look to the hard copy of the United States Code Annotated (USCA) of the relevant statutory sections to find relevant cases interpreting the statutes and then something like Findlaw or Cornell to look up the full text of the cases referenced. Every physical law library and many university libraries and public libraries without law schools affiliated with them will at least have a hard copy of USCA.
USCA will also note the West Digest System key numbers of the cases annotated, and you can then look at the hard copy of the West Digest to find other cases (possibly arising under different statutes) that implicate the issue that the West Key Number in Question references. A West Key number has a word or words indicating where the entry appears in alphabetical order in the digest, and a key number which indicates which section number within the word topic is addressed by the annotation. A full set of West Digests (which is a comprehensive compilation of all case annotations of all cases reported by the West Publishing Company, which is the most comprehensive of the annotation systems) would be in every state supreme court library and in every law school library, but you'd probably want to consult of university librarian if your library doesn't have one to see where you can find it.
Black's Law Dictionary is something you should just buy in hard copy and use as an unofficial textbook for the course if you are doing anything more than a nominal amount of legal research. Knowing the technical sense definition of terms of art is absolutely key in the medical sciences and in that respect, law is no different.
There are also a variety of not very expensive handbooks on legal research for aspiring law students (typically first year law students), any of which is adequate for your needs.
Another secondary source you should consider is a legal treatise in a subject-matter of law that you are working on. This is a topic area reference written by a renowned expert in the subject area that references all relevant legal authorities including statutes, regulations, case law, and other guidance (as well as legislative history) all in one place along with the author's own commentary. In health care, there may be legal treatises on health care law and there may be legal treatises on administrative law. Again, a library copy is one option, but in the case of a treatise an interlibrary loan is sometimes possible as well. The links in this paragraph are from Georgetown Law School and its publicly available online law library resources are particularly good.
I was hoping someone knew of any resources that mirror some of my
favorite citation/reference searches. This includes:
No freely available resources of this kind exist. Th closest you will get is Google Scholar.
The Health Care Law Professors Blog, which is searchable, is also a rich lode of resources. Many law review articles have open access preprints posted at the Social Science Research Network, which I've often downloaded paper from but don't know how to search.
The University of Michigan Law library has the best collection in the nation of non-U.S. legal texts in the original languages, but it doesn't sound like you'll be needing those.