I have been using the search function in the forums and the How-To guides and I can't seem to find an answer basically to the OP's original question. That is, how do we move between devices and what questions do we need to ask ourselves to result in a functioning device? This would be an excellent How To Guide.
For instance, I have an S4 mini on sprint and it has a sim card. I have thought about getting a Tablet (like galaxy tab 3 or Nexus 7) or whatever. Some tablets say WiFi Only, Others say WiFi LTE advertised on Amazon. If it has LTE, it seems like I should be able to get that tablet and put my Sim card in the Tablet to use mobile data, but I don't think that is actually how it works. It seems like there are 4 or 5 criteria that need to be met. First, Are both devices on the same type of technology (hardware in device) CDMA Vs GSM. Two, does the Carrier Lock the device(For instance for sprint, you have to buy a sprint phone or tablet, it specifically needs to state it is sprint compatible for it to work but for ATT it seems they are flexible, and you don't need a device specifically related to ATT, it just needs the right hardware). Furthermore, even if I get a sprint tablet, will my sim card automatically work in the tablet or do I need to go to sprint and tell them I want to add a device (i.e. does the carrier even allow you to swap the sim card).
There are probably more questions that need to be answered before ensuring sim swappability between devices.