What's the proper way to go about replacing this faucet fixture? Is it always replace every part at once or can I mix and match if I like part of it?
Unfortunately, Pull-Outs work very differently and can't just screw-on to the old faucet...anywhere in your case. Their sprayer's quite long hose must nest down into the sink cabinet or the wall's void.

Although, if you can remove the handle caps and unscrew the handles to take them to the store, then you may find a new faucet setup that lets you use the old handles on the new faucet.

Lastly, you may also keep the perfectly fine faucet and plumb-in a separate sprayer installed in the wall next to your faucet or on the countertop. This is a bit involved and a sprayer and hose intended/rated for constant pressure is needed...like a Pot-Filler's separate setup.

A Plumber would be best to use for that last bit and they can renew washers and whatnot in the old faucet to ensure a decade or more of continued admiration.