Pop out the old sim card. Put it in the new phone. That's it. It's POSSIBLE that he's reluctant to give out a bunch of information because there's no need for the information, and if someone is asking you for all that information, it's likely not for a legitimate reason? Please note I said POSSIBLE, not probable. I can only say that if my wife said she needed to know how to login to make changes to our mobile plan, I'd be a bit suspicious...but would gladly show her how to eject the sim card.
Note, the sim card you have could be larger than what the new phone takes. Unless the sim card is a decade old, it can PROBABLY be cut down to size. If you have a tech savvy individual in the family, it's not difficult (I personally have a sim cutter, but have accomplished the same with a pair of scissors and a template). You can also take it to a mobile store, it's not a hard thing to do. There's a good chance that, even if it's a larger size than the new phone takes, it can be "punched out" to the smaller size. A lot of sims come as a 3-in-1, where you can punch out the largest size, the next smallest size, or the smallest (nano) size.
If you have an iPhone 5 or newer, disregard everything I wrote about sim sizes...it already takes the smallest nano size, so you're good to go.