There are two ways of sending a group message, and most texting apps have a setting in Settings (in the app). You can send individual texts, one to each member of the group, or you can send an MMS to each member of the group - they each get a link to a text file that the provider stores on their MMS server. Each recipient downloads the same file.
If you have your MMS settings wrong in your APN (Settings/Connections/More networks/Mobile networks/Access Point Names - then the carrier), you won't get or send MMS, but the internet will still work. (Each carrier has at least one set of settings, so search apn carrier-name, where carrier-name is your carrier (like apn att or apn verizon). If they have more than one, go to the store and have them set yours to the proper one for your type of account.
Or set your texting app to send individual messages (or not send a single group message - some word it one way, some the other).
As to why your wife's G3 works and your G4 doesn't ... that's something both phones have to be looked at to determine. If your APN (the MMS part, around the bottom half of the information) and hers are the same for MMS, I'd suspect what's called your account provisioning - they set your account up wrong. A call to customer service (with most carriers - some hire less capable people than others) usually has them reprovision your account in a minute or two. (If it's Straight Talk, use their Facebook page - the people there seem to know more than the people at the customer service phone number. If it's Verizon, good luck. Their "customer service representative creation machine" has been broken for years - you know more about phones than they do. If it's AT&T, try your local store - if they can't figure it out, ask where the Device Support Center is. It could be in that store or in a nearby one. Their normal method of operation is fix the phone or replace it, but you walk out with a working phone.
Other carriers run the gamut from AT&T's "satisfied customer" policy to "we sold you the phone, got your money and don't care to waste any more time with you". There's a reason some carriers are so cheap.