Build a matchup-aware mulligan
Keep starting cards that help your first few turns. A cheap card is useful only when it has a job to do.
Choose a useful start
The mulligan lets you replace cards in your starting hand. Think about what you need to do in the first few turns.
A cheap minion that works on its own may be a good keep. A cheap spell with no likely target may not be. A more costly answer can be worth keeping if it stops a key threat.
Look at the whole hand. A card that needs two missing cards is a risky keep. If you already have the cards that help it, the choice may be different.
Think about the other deck
Against a fast deck, you may need early removal or defense. Against a slow deck, you may want a strong threat or a card that keeps making value.
If you do not know the deck, favor cards with several uses. Update your guess when you see more cards.
Going second gives you The Coin. Plan at least your next two turns to see how it helps.
Use data with care
A card's mulligan win rate is a clue. It can depend on the deck, patch, matchup, and other cards in hand. It does not mean you should always keep that card.
Try this sentence: “I keep this because it helps me do ___ by turn ___.” After the game, judge the choice using what you knew at the start. A good keep can still have bad draws.