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Loosen the tuning pegs all the way down, then remove the bridge plate to pull out the old strings - makes the whole job way easier than trying to work around them. Feed each new string through the bridge block, pull it up to the tuning post with a bit of slack, and wrap it around the post two or three times before tightening. Keep tension even on each wrap so the string doesn't slip, and tune up gradually rather than cranking it all at once. A string winder speeds things up, but honestly you can do it fine by hand if you're patient.
The bridge plate removal thing definitely speeds things up, but here's what makes the whole process smoother: pull just enough slack in each string so you can work the bridge plate off without fully loosening the tuning pegs. You keep some tension, which means the headstock stays more stable and you're not wrestling with a completely floppy neck. Once the plate is out, *then* you can fully unwind. Feed the new strings through the bridge holes, seat them properly so they're not kinked, and work your way up to the headstock.
When you're at the tuning pegs, here's the hack nobody talks about - wrap the string around the post only twice instead of the standard three or four wraps. I know that sounds wrong, but two clean wraps holds just fine and the string settles into pitch way faster because there's less material bunched up. Just make sure those two wraps are neat and the string goes under itself as it wraps down toward the post base. Tug the string a few times before tuning to seat it, then tune up. The whole job takes maybe 15 minutes instead of 20, and your tuning stays stable.
Stretch the strings out manually once they're close to pitch - grab each one between the bridge and the first fret and pull gently. Do this a couple times and the strings won't keep going flat on you every five minutes like they do when you skip this step.
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