Depends on what type of racing you are interested in - road race, circle, oval, pavement, dirt, drift, off-road, enduro, rally, etc.?
Most professional racers get their start in some race related field - most common is Karts. They are inexpensive to get into and can be dangerously addictive. They are generally shorter tracks, but many have started opening up to longer venues - opens doors to enduro, circle/oval track, and spring racing events.
Get involved in your local SCCA/NASA events. Join a club racing group - many have open track days to get you on the track at reduced / group rates. These are the stepping stones to get into serious racing. You;ll get exposure and experience in racing around others, familiarity with tracks, get comfortable racing the clock, racing others. See other racer's setups, see what sort of money is involved.
Can also get involved from the mechanical side as well. Working in a pit crew, volunteering time at a SCCA / Auto-X event.
In any event - you'll need a competitive license before anything takes off for you. Example, NASA (National Auto Sports Association) has several stages that you have to complete, generally you will not be eligible for a competitive license until you reach NASA HPDE 4 level.
Don't need to be a millionaire, but it isn't what I'd call "cheap" either. Novice licenses generally run from a $100+ - track time, materials, hotel stays, food, equipment, etc. Might consider renting a track car to concentrate on racing and not trying to beef up your existing car. Depending on the vehicle, could be from a few hundred per event to several thousands of dollars.
If you have a couple of thousand dollars burning in your pocket, I'd highly recommend a performance driving course such as Skip Barber Racing School or Bob Bondurant Racing School. Not only will you learn alot from those driving instructors in purpose setup vehicles, time spent there can qualify toward competitive licensing requirements.