> Public transit needs a lot of money and time
Public transport is far, far more cost effective than car infrastructure. And that's just direct costs - not even including the cost of sprawl (which makes all other infrastructure more expensive), road deaths and injuries, noise, pollution, storage costs for vehicles, the health costs of inactivity and social isolation, etc etc.
> build good transit for the busiest part of the city (downtown) and build large parking lots around the terminals
This is a terrible idea because the numbers simply do not stack up. A typical metro train can carry roughly 1000 people. A large car park might fill half of a single train. At a station with good frequency, a train will leave the station roughly every 5 minutes.
A much better idea is to run good regular public transport to the station, build bike paths to the station and quality bike parking at the station, and build more housing at/near the station instead of a big parking lot.
Parking lots near stations make sense only at the farthest out place. They are for farmers coming into town for that big once a month trip, and hobby farmers driving from their "acrage" to their day job. The vast majority of people should be taking transit from their door to where they are going.
Note that I said "place" not station - stations should be your highest demand places since they are so easy to get to. That real estate should be far too valuable to stores too waste on a parking lot. That parking lot should have a shuttle to the station, not be a station itself.
Remember once somebody has got into a car they have paid most of the costs of having a car. They will always be asking why not drive all the way instead of stopping part way. Your goal should be every family sells a car because they don't need two (they still keep a "truck" for towing the boat or whatever they think they need it for, they just don't use it for most trips and don't need a backup vehicle)