Replacing entire drivetrains on fixed gears is usually pretty straightforward. The main point of compatibility is making sure you're using a 3/32" compatible ring and cog if for some reason you use a 3/32 chain. In the more normal and advisable circumstance of using a 1/8 chain, it's hard to go wrong from a compatibility standpoint.
Unless there's something wrong with your cranks or you want something fancier, the normal thing to do would just be replace your chainring. If you are replacing your crank, you want any track-ish crank that's strikes your fancy, the nicer and more expensive (generally) the better for concentricity, plus a bottom bracket of it's manufacturer-prescribed spindle length, which will be spec'd to put the front chainline at the standard for track/fixed gear bikes, around 42mm. If you do decide just to replace your chainring, get anything of the right BCD and the tooth count you prefer that's made for singlespeed use. I've become extremely partial to the Surly stainless steel rings because they're well made, beautiful, strong, and economical for how long they last. If you're in the US, most of the house brand rings you'll encounter are totally fine too (Origin8, All-City, Dimension, etc.)
For the bottom bracket, if you stay with the existing cranks and presuming the chainline on the existing bike is good (the plane of the ring teeth and the plane of the cog teeth are within around 1mm), all you have to do is match your current spindle length.
Cogs are a place not to skimp ever. In the early days of the fixed gear revival and in the years before that, cheap stamped track cogs that are prone to stripping threads and in the worst case destroying hubs were fairly common. They're still out there, but a lot of us have been taught the hard way to steer clear, and thankfully the good options now outnumber the bad. Of the commonly seen good ones I would count Shimano, Surly, EAI, Origin8, and All-City (there are more but I'm speaking from what I've used/sold). I honestly don't think it matters too much which you pick once you're out of the junky price bracket, but I'm partial to the Shimano Dura-Ace cogs because of their increasingly rare confluence of exquisite precision and humble, utilitarian finish, plus they're quiet and last a long time. Lockrings are pretty much indefinitely re-usable.
For chain, the choice is basically use a basic, cheap, and functional singlespeed chain like the KMC 410, SRAM PC1, etc, a dedicated track chain, or a BMX chain if you want to get crazy. There is little to no downside in going basic. The other options have cachet value, and the nice track chains can be a little quieter perhaps.
It should be noted that some would answer your entire question from a perspective of what's quietest in each category, and that's not entirely wrong because it is an area where there are real differences in singlespeed drivetrain parts, particularly cogs.
