The Bite-Sized series of articles is meant to take the most powerful cards in Dominion and give a short explanation of what makes them powerful.
At the start of the game, tournament is already quite strong. Cantrip +$1 has an economic impact similar to Silver, and as your deck improves, it gets better than that fairly quickly. Additionally, compared to a pure payload card like a silver, the cantrip nature is much better for deck control, letting you more easily start chaining cards together..
Obviously, the other text on the card is important, too. Note that turning off your opponent’s Tournaments by revealing a Province can be a very big deal – often even bigger than gaining Prizes. If they would draw 3 tournaments in their turn, then by having a province in hand, you’re denying them drawing 3 cards, plus 3 coins, and potentially more than that, as the lack of ability to draw also increases the chances that their turn stalls out.
The prizes are also strong. Generally, Trusty Steed, Princess, and Followers are the strongest, and one of them is almost always taken first. Which of them is best is board-dependent – usually, look for what the rest of the board is missing (village, +buy/payload for many-buys deck, discard attack). Taking other prizes – including duchy – is usually a consolation or an endgame play.
Because of the importance of Provinces on both these axes, getting to $8 faster is generally more important than normal.