I specifically have a Trial of Solidarity that is only active "until end of turn." I am wondering for this and other relative cards, whether they are to be discarded at end of turn since they are used up, or if they remain as permanents with no use? It is especially important for this card, since it has an additional effect that depends on another card's entering the battlefield AFTER its been played.
An enchantment is a permanent, and permanents remain on the battlefield indefinitely. Only another effect can move them to another zone.
110.4. There are five permanent types: artifact, creature, enchantment, land, and planeswalker.
110.1. A permanent is a card or token on the battlefield. A permanent remains on the battlefield indefinitely. A card or token becomes a permanent as it enters the battlefield and it stops being a permanent as it’s moved to another zone by an effect or rule.
You are correct that once a Trial has entered the battlefield and done its thing, it's practically useless. When you play a Cartouche and return the Trial to your hand, you can cast the Trial and get its effect again. You can play the Cartouche the same turn or any number of turns later. The Trial will remain on the battlefield until something happens to it.
The idea behind the "Trials" card cycle is that you get a one-shot effect that you can reuse with Cartouches, with the downside being that the combo can be broken by e.g. destroying the Trials while it's sitting uselessly on the battlefield.
Trial of Solidarity has two abilities that are both triggered abilities. The first one triggers when the Trial enters the battlefield, and creates an effect that lasts until the end of turn, and then other triggers when a different card enters the battlefield, and returns the Trial to its owners hand. In the first ability, what lasts until end of turn is not the permanent itself, but the effect created by that ability.