united-states
This is challenging to do in most state and local trial courts, because full trial court transcripts are usually only made available to the parties to a case being appealed and appellate judges, and because state trial court filings are often hard to access.
Most non-sealed documents in state and local trial courts in non-juvenile cases are a matter of public record and can be obtained upon request from the clerk of the relevant court (lawyers can look directly through the state e-filing system, but sometimes have to pay a fee to do so), although you typically have to go in person to request them and don't have an index of records available to you to review before asking for them. There is also typically a research and copying fee to have the clerk of the court provide them for you.
But the PACER system of the federal courts makes all non-sealed court filings available to the general public, including motions and briefs that articulate the legal reasoning of the lawyers in writing. Oral in court proceedings in U.S. District Courts, however, are usually available only by being present in person to watch them.
U.S. Supreme Court oral arguments are available in audio form (but without video) shortly after they are made.
I don't know the practice in other U.S. states, but in Colorado, all oral arguments in the Colorado Court of Appeals and Colorado Supreme Court are live streamed in audio-visual format and can be streamed after the fact as well.
You can also just walk into a courtroom for almost any court case and sit in the gallery and watch the proceedings. This is true for almost every single kind of court case except juvenile cases, trade secret cases, and national security cases. There is rarely a shortage of space, or a line you have to wait in, to do this, except for trial of the century type criminal cases and for the U.S. Supreme Court in Washington D.C. This isn't done very often now, but in the 19th century and early 20th century, before television was invented, a surprising number of people did this for entertainment. These days, the privilege of doings so is exercised mostly by journalists, by students interested in law, by friends and families of the parties, and by people waiting until their own hearings come up. You have to identify yourself, because people who have to testify later in the case are excluded from the proceedings, but this is usually no big deal.
Sometimes it is possible to observe live streams of state trial court proceedings in Colorado via WebEx, with each courtroom having a link, but again, with the requirement that you identify yourself. These are not available for streaming after the fact.