This step is one of those things you kind of have to feel your way through over time to see what is the best fit for you, taking all things into account, such as how much time you have for home brew stuff, how that time is distributed, availability of equipment and space, your patience level, etc.
I say this because I've tried all manner of approaches and they have all worked, with some of them being a bit more error-prone than others.
Easily the most fool-proof method is the so-called "low and slow" method where you put the keg at serving temperature and pressure and let it ride for 1-2 weeks. Pretty much the only "risk" of this method is if you have an undetected leak in your CO2 system you will come down one day to an empty CO2 tank, which is rather infuriating (ask me how I know.) Check your connections and seals carefully.
If you search this forum you can find some pretty specific instructions on the quickie-carb method of chilling the keg, hitting it with high pressure, and rocking it on the floor. That method definitely works and I have used it in a pinch when I have needed to get kegs on tap quickly for planned events, etc. The drawback of course is that it's not difficult to overshoot the target and end up with over-carbed beer (ask me how I know.) Also, all that rocking will take every last particle of sediment in the beer and toss it back into suspension, so your beer will be rather turbid and ugly until that has a chance to settle down again.
Then there is the middle-ground type method where you chill the keg and hit it with higher-than-serving pressure for a much shorter time than low-and-slow; i.e., 18-24 hours or whatever it works out to be. There was an article on Brulosophy that provided some guidelines on this and I recall following that process with success (although I haven't used it in ages). I should note that I have also accidentally over-carbed using this method. Not a huge deal to fix but a PITA.
These days I usually just use the chill-and-rock method but I'm very conservative with it, using about 25psi for 45 seconds or so. I'm not trying to fully carb the beer, I just want a headstart on it. Then, I put the keg on my serving system to finish the process. Once there, it gets topped up on CO2 and is ready in a couple days, but the risk of overshooting is pretty much nil, and those few days gives the keg a chance to settle a bit as well.