Wow Ivan, you have some guts if you want to start grid racing straight up...It's a pretty nerve racking experience first time off..Not sure you are understanding the difference between proper door to door racing or just peddling a car around the track as fast as you can go!
There are club level race series that you would be able to get involve in....The main cost like the guys are saying is your car..
There is also licencing, logbook, racing apparel and most door to door racing requires the car to have a full roll cage so would be hard to have a steet driven car that you race..I reckon if that was the way you wanted to go
Maybe check out a few race and go kart days and have a chat to people/entrants and see if you can make your mind up which way to go..
Even super sprinting a car is loads of fun without the stress of guarding your position and line.
It's an expensive hobby but I wouldnt have it any other way