Powerups from Galleria/Secret Stuff
Only buy the turbo fill powerup, which is affordable after you beat any Cup within the Mega Carrot Tourney. https://imgur.com/a/BnDrqg2 The other gags aren't worth picking up, and would (probably) reduce the odds of picking up clovers and turbo fills.
Boosting
Normal laps (not final lap, no points awarded for the lap): Boost as soon as you fill your turbo and there is a semi-straightaway.
Final Lap: Remember about where is the last place you could boost and be able to grab enough turbo cans before the final finish line. Once you pass that area, do not boost until the boost would carry you over the finish line. This is to prevent chance gags from hitting you after your final boost. Save that final boost for if you do get hit by a gag, or someone passes you with a boost of their own.
General boost tips: Be careful of spending a turbo too soon. Often times you boost, then a gag hits you after and you have no way to accelerate quickly anymore. If you're way out front, boost as often as possible, following the above lap-specific guidelines. If you're in the middle or back of the pack, wait until it's safe to boost (watch the racers in front of you for equipped gags).
Acme Pickups
If you're in the middle/back of the pack: Pick up and use everything immediately. If you can't hit someone immediately with a close-ranged gag, drop it immediately (square button on old PS2 controller). Never save damage gags if you're surrounded by racers or if you're behind everyone.
If you're in the front 3: Prioritize damage gags and deploy them immediately towards your 1st-3rd place rivals. If you get the turbo fill powerup or the clover, generally you should use them immediately to protect you or break out of the pack. The exception is when you're sitting in 3rd and watching the front 2 racers pummel each other with gags. Wait until they don't have a gag equipped to break out front.
If you're in 1st: Drop every damage gag immediately, with the possible exception of fast-use seeker items like the stormcloud or the rockets. Remember though, while the gag animation is playing, you can't boost or pick up new gags. Most of the time, it's best to immediately drop gags.
The only gag you should care about is the four-leaf-clover, and the turbo boost fill (unlockable mid-way through the game). Turbo fill is self-explanatory; wait to use it until right after you boost so that you get a full fill. Save clovers for as long as possible. Once a seeker gag like the rockets or the elephant approaches, wait for a small time and then deploy the clover. This is to allow the clover to remain in place as long as possible after protecting you from the immediate gag danger.
Race-Type-Specific
Normal races (Acme Pickups + turbo cans): Always prioritize turbo cans. Always. Miss Acme pickups if it means a chance to align with a row of turbo cans coming up.
Gag-only (no boost cans): Follow the general gag use guidelines above.
Boost-only (no gags): The speedrunner's dream race. Spam turbo as often as possible. Keep in mind the points in the track where boosting will send you over the side of the track, however.
Griefing computer players
Always try to stay out in front. When you are ahead of any racer, try to pick up the boost cans and acme pickups which are most easily obtainable. For example, on tracks with tight turns, some pickups or boost cans aren't easy to grab unless you slow down a lot. By taking the easy grabs, you're depriving the computer of chances to overturn your lead. This is especially important for boost cans. Be aware that the computer can do this to you on normal difficulty (looking at you bugs bunny).
In the final tournament, there's a race where the only gag are fake, exploding acme pickups. If you practice the boosting guidelines above, you should have no problem staying near the front or in 1st. To really cement your lead, drop any fake acme pickups you get, directly over turbo cans whenever possible. This will hurt computer players trying to catch up with boosting.
Advanced fall gag evasion
L2 and R2 (PS2 controller layout) are your friend. These allow you to bump yourself to the left or right, which can save you from falling if you time it right. Also, if you hit it precisely, you can bump out of the shadow of a falling object, but it has to be right as it's about to hit you.
The main way to avoid falling gags (besides using a clover), is to take advantage of the raised sides of the track. If you cuddle up in the raised sides just right, the shadow of the falling gag is deflected and it will miss you. Be careful of trying this around curves, in which case you often have to swerve in and out of the side to stay on the track. If this happens, the falling gag can hit you if you aren't swerved over the side at just the right moment. This technique is most easily used inside of tunnels with hard sides, which prevent falling off the track. Keep in mind that running over the sides and running against the walls will slow you down (though not as must as a falling gag).