If its an iphone the new phone (if factory reset) guides you through the transfer. Assuming you have your old phone backed up to icloud, you put the phones next to each other and it uses near-field communication to transfer initial settings such as wifi, account etc. It asks a few questions but gets the job done easily, the icloud restore can take time to get lower priority items (eg photos) but if everything is backed up everything comes across including contacts, old texts, old emails, apps etc. Some apps pull their full settings across, some more secure apps don't (eg banks, post office id etc) so you need to log into those again and enable fingeprint/face recognition etc.
If you have limited mobile bandwidth, make sure you are in wifi range.
Biggest disappointment with me was getting a new provider to transfer my old number in, this should be (and in the past has been) simple but it took 3 months to iron out problems recently, I could receive and send texts and make calls but couldn't receive calls when on a mobile signal (fortunately I could receive calls through wifi). The problem had been obfuscated by a faulty mobile mast and a slight fault on a phone at the same as the routing fault, three separate faults which together made no sense.