There are a lot of directions you can go with this. One of your biggest expenses is going to be paint.
Groody Bros have a good reputation for that.
Some of the other bits can be a rabbit hole. Brifters or downtube shifters? Wheel size? Number of cogs in the rear? Upgrade the brakes? A lot of these upgrades are tied together so if you change one you need to change others. Brifters (usually) means changing derailleurs for ideal gear changes. I’m not sure what is on your Schwinn but changing from 27” wheels to 700c may require brake changes. If you plan to run fenders/mudguards you need to check if those will fit or can be modified to fit.
I use a lot of qualifiers for my statements because each build presents it’s own challenges but if you are patient almost anything can be massaged to work.
Good luck!