It depends on the type of game you (or your DM) want to run. In a simplified game (which seems very popular for 5E), it's just another type of coin with a straight up exchange rate. In a simulationist game, characters have to determine the value of the gem/jewelry and then haggle to get the best price. There are many steps in between, based on the desire of the DM.
As a DM I give gems, jewelry, and art objects a quality modifier (ranging from -90% to +400%, with most having no modifier). Characters can evaluate a number of items equal to twice their Int modifier (min 1) during a Short Rest (similar to identifying magic items). I also give the buyer an economic modifier (from -50% to +25%, with most having -10%). Except for large cities, there isn't more than one buyer in an area so there isn't any haggling. If there are multiple buyers, the seller can attempt to haggle (Cha/Persuasion check), but once this is made, the player cannot back out of the deal.
As a player, I try to keep as many gems & jewelry as I can. They are lighter than their coin equivalent, so they allow for more treasure to be carried. I like to keep art objects too, but this depends on the campaign. If we're travelers, the size and weight of the art object makes a big difference. If we have a base of operations I can keep my stuff, I like to decorate
