Since one's immediate instincts about how to play Tringo are dead wrong, I'll give a quick overview of basic strategy:
- If you can form a 2x3 rectangle, take it
- If you can't form a 2x3 rectangle, do not jump to form a 2x2 rectangle, only do that if you are in danger of not being able to place your next piece.
- When placing pieces without forming rectangles, try to keep your pieces lined up against the edges, and try to avoid enclosing empty squares so that very few shapes can fit into them.
I believe that with sufficiently skilled play one can consistently get over 200 points.
The theory here is that you're trying to get as many points per square placed as possible. 2x2 squares only get you 1.2 points/square, while 3x2 rectangles get you 2.5. You can get more from 3x3 rectangles (3.33) and obviously you should take those if you can, but those occur so rarely that they don't impact strategy much.
My suggested improvements to Tringo are to make 2x4 rectangles be worth 25 points (getting rid of them as two 2x2s severely penalizes good play, while making them worth only 20 would give them the same per-square point value as 2x3s, even though they're much harder to make) and add a pieces left countdown so you can see how many pieces are left towards the end, since that impacts strategy a lot.
When there are many people playing bingo-style it of course makes sense to be a lot more aggressive, mostly in the form of trying to avoid 2x2s even more than normal, since a very high score is required to actually win. Since Tringo is mostly a game of skill that doesn't completely change the game though.
Strategy aside, the story of Tringo is quite an interesting one.