What are you using to "swap"? I did this recently with a samsung, using smart switch. It should work a similar way. You have to have the swapping app on both phones, then put the sim and memory card (if there is one) in the new phone and then run swap on the new phone. If it works like smart switch it should tell you what to do from there.
What is the make and model of phone you are trying to open? Most of them do not open these days they have small drawer with the sim card in it that pings out with a paperclip.
I know 'how' to open the old phone, but it just won't open. It's an LG Joy.
I am going to forget about this until I feel more alert. I am prone to making simple issues turn into disasters when doing stuff without my brain powered up.
Just be aware that your new phone might take a different size SIM from your old one - SIMS have got smaller with time. If that is the case your network provided will give/sell you a replacement SIM which has the same number as your old one.
The LG Joy takes a micro SIM. Most modern phones take a nano SIM. If this is your situation you just need to go to you service provider and get them to swap the SIM for a smaller one. If both phones take the same size SIM, then as others have said, just pop it out of one and put it in the other. The SIM is effectively your phone number.