If you cast Resurgence when it's not a main phase (perhaps with Omnispell Adept's ability), creatures you control gain first strike and vigilance, but there's no additional combat phase or main phase. If you cast it during an opponent's main phase, there are additional combat and main phases, but that player attacks during that combat phase, not you.

Resurgence affects only creatures you control at the time it resolves. Creatures that come under your control later in the turn won't gain first strike and vigilance.

Split cards have two card faces on a single card. The characteristics of the half you didn't cast are ignored while the spell is on the stack.

If you copy a spell that's half of a split card, the copy copies that same half. For example, if you copy Assault, the copy is also Assault, not Battery.

To cast a split card, choose one of its halves to cast. There's no way to cast both halves of this split card.

Each split card has two names. If an effect instructs you to choose a card name, you may choose one of those names, but not both.

A split card's characteristics are a combination of its two halves while it is not on the stack. For example, Assault // Battery has a mana value of 5 while it is in your library. If an effect allows you to search your library for a card with mana value 4 or less, you can't find Assault // Battery.

Each split card is a single card. For example, if you discard a split card, you've discarded one card, not two. If an effect counts the number of sorcery cards in your graveyard, Assault // Battery counts once, not twice.

If an effect allows you to cast a spell with certain characteristics, consider only the characteristics of the half you're casting. For example, if an effect allows you to cast a sorcery spell with mana value 2 or less from among cards in your graveyard, you could cast Assault this way, but not Battery.