If you cast a card "without paying its mana cost," you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the card has any mandatory additional costs, you must pay those to cast the card.

If the card has {X} in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

You cast the cards one at a time, choosing modes, targets, and so on. The last card you cast will be the first one to resolve.

When casting an instant or sorcery card this way, ignore timing restrictions based on the card's type. Other timing restrictions, such as "Cast [this card] only during combat," must be followed.

If you can't cast a card, perhaps because there are no legal targets available, or if you choose not to cast it, it will be put into your graveyard.