When you're racing, better suspension > better brakes. Who needs to stop (or brake pedals) when you can cut through corners with ease?
But in honesty, blanks and pads with stainless steel brake lines and a DOT4 brake fluid will perform very well on the track. That goes for around $450-$600. Then you can save up for stickier tires and light weight wheels (~$3000). This setup will be just enough for a good weekend track car.
Bigger brake systems are really intended for wheels/tires with larger diameter and larger 'moment of inertia'. The Civics have puny wheels/tires. It will stop a car quicker than a normal sized brake system. But you'll also need the right tires, maybe fatter and stickier tires that will produce a bigger surface contact in order to stop as short of a distance as possible.
Using the 2006-2011 Civic Si as a reference, their K-sport Big Brake Kit goes for about $3421 ($1663, for the 13" fronts with 8-piston calipers and $1758, for the 11" rears with 4-piston calipers). Then you'll end up buying ~$3000 wheels/tires. That will be around $6500 just on the BBK, brake fluid, light weight wheels, and beefy sticky tires.
And in addition, if you get a rear big brake system, the OEM handbrake will NOT work. Lol. Not with 4-piston calipers holding on a rotor.