I always change the 3 bass side strings first, then the three treble side strings. No reason, it's just the way I've always done it. I love playing on new strings and could never wait 24 hours. In fact I will change someone else's strings for them if I can play the new strings for the first 20 minutes.
I fasten the strings at the bridge or tailpiece, then pull them tight around the tuner post, wrap twice and through the hole. Then I use a string winder to bring them close to pitch before fine tuning. I like to stretch them by pulling them away from the fingerboard.
I always try to clip thestrings, but if I find myself changing a string without a tool to cut it, I will take a loonie (Canadian dollar coin) or a quarter and use it to curl the string (like my mom used to curl Christmas present ribbon with scissors).