I'll preface this response with the disclaimer that I'm not a lawyer and that this isn't legal advice. Depending on your jurisdiction, the amount you'd be seeking in damages would likely determine the type of case (e.g., small claims vs. civil case) and whether it'd make sense to retain an attorney or whether to prepare/argue your own claims, filings, proceedings, etc., and how robust this type of evidence needs to be to be useful. Before it gets too far, it'd probably make sense to contact an attorney (maybe one who offers a free initial consultation) to get qualified legal advice, particularly as it relates to the specifics of your circumstances.

That said, I would think that using publicly available site archival services, including the Wayback Machine available at archive.org, in addition to the sites others are linking here, would be a good idea, as well as possibly taking a video of yourself accessing the relevant page(s). Basically, having proof that the company advertised a particular aspect to their service that it wouldn't be able to hide simply by later removing it from their website later might prove helpful. I believe I've heard that companies can request for archived pages to be removed, so having archives across multiple sites (and a video that is completely within your control) would make it more difficult for the company to pretend that they never advertised those processes, assuming they're not already written down in any other agreements between you and the company in the first place.