For indoor carpet you'll want to invest in a gear diff first.
For a tight budget, I would consider the
Radiolink RC4G
It's not highest quality radio, but I use it as a backup and I still get the same fast laps and consistency as I get with my high end system.
If you want a middle grade system, then consider a used Futaba 4PLS which includes steering speed adjustment, that feature is the single most important feature in preventing traction rolls in high traction conditions. I would pass on any system that doesn't offer this feature.
For entry level mod, I would start out with a 7.5T motor, I actually have a brand new sealed Trinity 7.5T motor that I'm selling for $40 here:
Bill's Stuff For Sale
I have tried many brands of ESC over the years, of the low budget stuff I would consider the
X-Car 120A ESC which will get you started, plan to replace this ESC with a higher quality
HobbyWing V3.1 120A ESC down the road, HobbyWing products will stand the test of time and they offer deep discounts to replace any damaged ESC's outside of warranty.
Be sure whichever servo you decide to get has replacement gear sets and is in the speed range of 0.08 to 0.12 sec/60°, this is where having the steering speed feature in the radio will help you fine tune on the track later.
I'm partial to the
EcoPower WP120T which has
replacement gear sets for only $13.
There are some budget brand servos with
JX Servos being my personal favorite, but consider this brand as disposable because they don't offer replacement gear sets where most servos I have raced with will tend to wear out the gears once every year or so. I have some servos that are 8 years old and have replaced the gears in them 2-3 times making the long term investment well worth it
Best of luck!