I don't own a car (and don't particularly want to). So I have to get to everywhere I want to by walking or using public transport. Here's what you would have to do in my small city to make people abandon their cars for walking/public transport. It'll cost way, way, way more than £10,000.
Make public transport really cheap, really frequent, really efficient... and really go everywhere people want it to. Buses/trams/tube trains/local trains which run every 5 or 10 minutes all day, every day and on every route to myriad destinations are what the public wants. They don't want to stand for long periods in the rain, with them, their children and their shopping getting soaked while they wait for a bus/train.
But because the bus companies are trying to maximise profits, they don't do this ultra frequent, ultra efficient timetabling. So there are buses which only run once an hour. Buses which don't run after 19.00. Buses which don't run on a Sunday or on public holidays. Buses which do run every 10 minutes but are so packed no more people can get on board. And Grud help you if you want to go from one side of the city to another - that'll meaning changing buses/trams in the city centre. Lucky you - you get to stand in the rain twice! Maybe 3 times if you live in a really big city. Then you get to do the same on the way home! :-)
Redesign your public transport. Parents with pushchairs want to use buses. People in wheelchairs want to use buses. People with luggage or big parcels want to use buses. Oh dear, 2 prams, 1 wheelchair user and 1 person with an enormous suitcase are waiting at the bus stop and the bus only has room for 2 of them, because of the way it is laid out. More waiting in the rain...
Redesign buses to have more room for families with prams, people carrying enormous stuff from Ikea, kids with scooters, etc. You'll have to sacrifice seating to do this, so you'll have to run MORE buses to compensate. Train and tram carriages should be the sort which can carry bicycles and lots and lots and lots of prams/wheelchairs. And have decent luggage racks, not the stupid little ones which barely fit a briefcase.
Completely redesign the layout of your city. All sorts of things in modern cities have been designed with cars in mind. Edge of town/out of town shopping centres, cinemas and/or industrial estates, for instance. When public transport is intregrated into these, it is usually as an afterthought: Ooh better ask the bus company if they want to run a bus to the new multiplex... Oh, they don't. Ah. well, never mind.
So don't let anyone build anything without incorporating a railway or tram line or tube line as part of it! Rail transport indicates you really mean it. Buses are a poor alternative.
Crash house prices and drop rents. People are commuting further and further by car, because they can't afford to live in or near the city centre. Housing estates are built on greenbelt land because it is cheap. If you make the centre and inner parts of the city affordable, people can walk or take a short bus ride to work.
Invest your $10,000 in a physics professor's research project and hope he invents teleportation. Public teleportation is pretty much the only way to sort out my city's transport problems!