The quickest way I've found to get some good starting money - not extremely rich, but a good starting sum - is to start out focusing completely on whims and aspirations - the goal is to get to 3000 points as quickly as possible. Switch around aspirations - there are several that have easy level 1 goals that can get you some quick points. At 3000 points, get the Connections reward. Then start joining and quitting careers. You'll start at level 4, picking up the awards for the previous levels - including the bonuses. That gets you several thousand dollars, and you can sell any of the career rewards you don't want.
Keep in mind, though, that if you later rejoin a career you quit, you'll start at level 3 the second time (I think you always start 1 level below where you were when you quit, but I haven't tested that extensively). So if you want to continue in a career, make that the last career you join for the specific sim and just keep going in it.
(By the way, if you do quit and rejoin, when you go from level 3 back to level 4, you won't get the career rewards again.)