So why does the client want you to do it if they know that you know almost nothing about databases? Better to get someone in who does.

From your requirements, there's little I can suggest.

The ability to take a backup - SQL has that built in. Native backups, maintenance plans. What kind of backups and how often depends on what the allowable data loss is, what the maintenance windows are, what's the max allowable time to restore and other factors.

Archive a database - can be as simple as back it up and restore under another name, or can be a complex custom archiving process.

Search a backup - not with the native backup tools. I don't offhand know of any 3rd party backup tools that allow full-blown searches.