To get the high-speed bonus you must stop the arrow on the green area of the half-circle speed bar, rather than the yellow or grey areas. Getting first bonus is important and it's the most difficult to get. When you get a perfect start the horse rears upwards on its hind legs.
To get the a perfect hit bonus you have to aim the lance so that it hits inside of the center of the circle. If it hits outside of the circle you'll get a lower rating from great or worse, depending how far you're off from the perimeter of the circle.
Purchase equipment using either gold coins or gemstones. Purchases of new equipment will finish immediately while upgrades will take time to complete. Each piece of equipment can be upgraded for a maximum of six times.
Rival Knights is free to play but you can pay real money to get coins and gemstones to upgrade your equipment.
Trumpets act as the scoring system for the tournament and your position on the global leaderboard. You receive trumpets for each match. The tougher the opponent you beat, the more trumpets you get. You will receive 5 trumpets for a loss or tie. You can add teammates from your allies list to help improve your trumpet total. Teammates give you a trumpet for every match they win. You can choose up to three teammates which, if they're active, could give you a big advantage in the event. You needn't choose a teammate in the group you happen to be in. There are multiple tournament groups playing simultaneously so pick three active players and revisit your list later if you find the trumpets received dwindling. On the main page (the one with "events" and "play" on it), there's a link to the leaderboard page. The left tab is the global leaderboard, the right is your ally leaderboard. You can track them there and use the ones who are most active as your teammates.
Even if you don't place in the top ten, you can still earn gems for participating. Plus it's a good way to make some coins to help fund your equipment upgrades.