EB05-021 Alvida — Your Priority
EB05-021 Alvida is your best card to keep.
With your Leader effect, you can play her for only 5 DON!! + one discarded card. Her On Play then lets you draw 2 cards and play a Cross Guild Character with a Cost of 8 or less from your hand.
Your most general line is:
Alvida → Dracule Mihawk OP09-048
This gives you two Characters with 7000 Power, a Blocker, and a refreshed hand thanks to the draw effects.
Keep Your Searchers
Mr.3 (Galdino) OP09-056 and Cross Guild OP09-057 are also excellent cards to keep.
Their main role early in the game is to find Alvida. Since both only look at the top 4 cards of your deck, you may need to search multiple times to find your key piece.
You can therefore keep multiple Searchers if you do not yet have Alvida in hand.
Dracule Mihawk — Set Up Your Alvida Target
If you already have EB05-021 Alvida, Dracule Mihawk OP09-048 becomes an excellent card to keep with her.
Mihawk immediately gives you a 7000-Power Blocker while letting you draw 2, then discard 1. He protects your Leader, improves your hand, and is a Character with a high enough Cost to help set up your future OP09-051 Buggy.
Alvida + Mihawk is therefore your most versatile opening combination.
Crocodile & Mihawk Against Removal Decks
Crocodile & Mihawk ST25-003 is especially worth keeping with Alvida when you know your opponent has Removal.
Its protection lets you discard a card instead when one of your Cross Guild Characters would be removed from the board by an opponent's effect.
In matchups where keeping your board is essential, Alvida → Crocodile & Mihawk may therefore be preferable to Alvida → Mihawk.
Do Not Look for OP09-051 Buggy Too Early
Even though OP09-051 Buggy is your main late-game threat, it is not your Mulligan priority.
Before you can really use him, you need to build a board with enough Characters that have a Cost of 5 or more. It is therefore better to look first for Alvida, your Searchers, and your first development targets.
You will then have several turns and draw effects to find Buggy when your board is ready for him.