How many links does a student of the Citadel need to obtain in order to be officially recognized as a Maester? Some like the Valyrian Steel link seem to be optional but I can’t imagine the Citadel providing a Lord with a Maester who doesn’t have, say, a gold or silver link.

Technically, as many as are needed to go around their neck, so some maesters have more links than others:

He fingered the collar of his order; a heavy chain worn tight around the neck beneath his robe, each link forged from a different metal…

His maester’s collar was no simple metal choker such as Luwin wore, but two dozen heavy chains wound together into a ponderous metal necklace that covered him from throat to breast. The links were forged of every metal known to man…

The old man touched the maester’s chain that hung loosely around his thin, fleshless neck…

Maester Colemon stood beside him, his chain of many metals hanging loose about his long, skinny neck…

Marwyn wore a chain of many metals around his bull’s neck. Save for that, he looked more like a dockside thug than a maester…

As to which are essential, I would say that black iron would be essential, since without that you’re not connected to the ravenry network and that’s a main reason why lords have maesters. Probably silver is also essential, because there aren’t really alternatives for trained medical care. 

But the others…yellow gold is perhaps not always necessary if one has a skilled steward (and indeed, I’ve never been quite clear on the division of labor between Maester Luwin and Steward Vayon Poole), iron is perhaps useful but it always seemed to me that military training came more from masters-at-arms and a lordling’s father, bronze or Valyrian steel seem definitely optional.