When you pick up a six-pack of cans, it's possible to park yourself close to a non-player character and rack up points by hitting them with the cans. The faster you are the better, so the beachball speed-up can be a big help here.
When the alligator comes to eat you (because you're taking too long to progress down the river) just hit him with one of the cans. This also works for certain other obstacles such as the floating mines (seeking or non-seeking). If you are quick enough, you can re-destroy the same mine several times and release multiple bonuses such as letters, treasures, sixpacks, etc. You can only throw one can at a time so any can already thrown will disappear once you press throw again.
Collect as many points as possible and try to push your opponent into obstacles. Swish up to five gates in a row and you will correspondingly advance your score multiplier (very important). Basic gates start at 2000, 1000, 500, 250. Once passed through the gates will decrease in points until they reach 100. Swishing a gate and then bumping it numerous times before your opponent can pass is an effective move. Pick-up cans (you can carry up to 9 at a time) or a sixpack (which gives you unlimited cans). The cans are used as "ammo" to hinder your opponent, stun other game characters, or to destroy certain obstacles (some obstacles are only temporarily destroyed). When you have a blowout you lose your sixpack and your score multiplier resets to 1. Your multiplier will go down each time you miss a gate.
Obstacles in the water (icebergs, bushes, mines, etc.) can reveal hidden points, sixpacks, patches (extra life), beachballs (speedup), including the fabled colorful letters. These letters are worth 10,000 points a piece. Try to have your score multiplier as high as possible to maximize the letter bonus at the end of each class. Be on the lookout for the 25,000 point gate since swishing it can give you up to 125,000 points if your score multiplier is 5! Unlike the other gates however, you only get one chance at the 25,000. Also be on the lookout for falling rocks, jumping penguins, submarines with nasty torpedoes, etc. On the first and second start gate you can utilize the warp whirlpool that will send you to a higher class. Many non-player characters chase you along the way. Watch out for the alligator, he likes to come out when you stall and he always comes from the top of the screen. You must go through a gate to repel that alligator before he appears or before he gets you.
A simple way to get a very high score : In only a minute or so, there's a way you can rack up over a million points. On Class 1, after going towards the first gate, direct your innertube to the right-hand side of the screen. You should see a sign that says 'This Way to Class 2' and a whirlpool should appear next to it. Shoot the twigs in front of the entrance and paddle back up. You will now be at the start of Class 2. Immediately move to the right of the screen, the same sign will appear, only this one warps you to Class 3. Shoot the iceberg in front, paddle back up and end up on class 3. The best part about the warps is that after each one, you get 600,000 points each. That's 1,200,000+ for doing very little work. An added bonus is that you will have four patches left in reserve thanks to the excess points you will have accumulated.