Most people doing their first game rinse-and-repeat the Sunday Cup a few times to build up a moderate amount of cash. The FR and similar series can be won by modifying until you overpower the competion, although for all except the 4WD very minor modifications are required if you have a good car to start with.
Buying a turbo Mitsubishi GTO and applying all mods might allow you to win the Mega-Speed event, which pays out relatively quickly. And in that state it would also stomp all over the 4WD series. (To be more fair, don't do the Cr85,000 "race modification").
The big money-maker is the Normal Car contest (for unmodified cars). A Dodge Viper or one of the TVRs cleans up there, once you learn to keep the slippery tires on the road. (About like "comfort" tires from GT4). Eventually, every Japanese manufacturer has a car you can win with, and real experts tell me you can win with a Chevrolet, too. Once you can win that series, money becomes almost no object. (One Normal Series equals one fully-modified car, basically).
And then there's the wonderful XXvsXX series. A race-modified version of a "normal" car can be fun there--maybe a GTO, or Viper or TVR. Actually, many non-race-modified cars can win too, e.g. a turbo GTO. Or it's fun to use a race-modified version of something pedestrian, like a Subaru Legacy Wagon or something. The great thing about these series is they are three series all having the same sets of races--the difference being the allowable entrants. So you get to learn those courses well. (Each series has a different character; in addition to the different line-ups because of the different countries allowed, the AI are faster/better in different particular races in each of the different series). If you win one of those series you'll get an "LM" prize car which can easily win any race it can enter. The model you get is one of two, randomly selected, so you'll want to race each of these several times to collect both from each series. (Two possible colours for each, too. Everyone seems to find some particular car/colour combination extra difficult to win, too--not always the same one for everybody).
Once you've finished most of the B and A license races (there's also the Clubman Cup and GT Cup in the Gran Turismo League; the prize from the GT Cup can actually win the JPvsXX series, but is difficult to tune and handle) you'll want to go for your IA license and try the GT World Cup, the endurance races and . . .
The Hard-Tuned Car World Speed Contest.
In many ways that seems like the all-round best series in any GT game ever. (Well, the fact you only face two possibly lineups sort of detracts a bit).