The quickest way is not always the quickest way. I've found - if you have the time to do the time-trials in arcade mode, you win the Esperante GTR1.
You can use it to win the 'Like the Wind' championship, which nets you a 787B.
From there you can use the 787B to run the Super-Speedway Enduro. It's a longer race, but the prizes are good, and you get a nice wad of cash.
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Another way I can think of is to run tracks in the 'Race of the Red Emblem' in Amateur mode - 3 laps for $30,000 each time. They are the highest paying short races in the game - in terms of $ per lap. You need a well tuned car, though. I used the Loctite Zexel GT-R - it's overkill, but it racked up the dollars quick enough.
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I haven't run GT3 for a little while (my PS2 is on the blink), but you'll find there are some reasonbly sized championships in Amateur mode (Like the European or Japanese cups) that have five x 3 lap races with decent prize money and a reasonable car. They are fairly quick and easy to win.
If you have time to do a championship where the Polyphony001 (or the equivalent F/xxx for NTSC) is a prize, they are worth the most money to sell - the Polyphony 001 is worth about 825,000 credits or so.
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Have a look through the car lists in the game guide section of this website, and check out what cars have what sell-value. If you are going to do championships, the higher value cars are the ones to go for.