No, unfortunately you will not be able to use a Sprint iPhone 4s on Verizon and you certainly will not be able to use it on AT&T. The iPhone 4s does have a sim card slot in it, but this is only for use with international travel so that you can use local sim cards in your phone when you travel, but when you return to the US you would need to use it on Sprint again and you will not be able to use it on AT&T - if you want to use AT&T, then you will have to buy a new phone that is a GSM phone and not a CDMA phone.
As for switching to Verizon, then unfortunately, you would need to buy a new phone specifically for use on the Verizon CDMA network.
For either of the things that you are asking, you need to buy a new phone whether it be another CDMA phone or a new GSM phone.
The situation is no different for an iPhone 5s either.