What’s the safest way to transfer a domain without breaking everything

I'm trying to move my domain from a registrar that constantly upsells me to one that's supposedly easier and more secure. All the guides say it's straightforward but then throw around terms like DNS, nameservers, DNSSEC, and email records like everyone just knows this stuff.

Honestly worried I'll click the wrong button during transfer and suddenly my site returns 404s, email stops working, and my small brand disappears for days while support tells me "just wait for propagation." Anyone have tips on how to handle domain transfers safely without everything falling apart? Would really appreciate any advice.