When MMS first started becoming a popular feature on phones, most of those phones had pretty crappy cameras that took very pics with very small file sizes...unfortunately the technology of cell phone cameras has far exceeded that of MMS. As noted, for example, Verizon's size limit of MMS messages is 1.2MB, and that is actually on the high end compared to many other carriers. The S4's 13MP camera, on the other hand, is capable of taking hi-res pics that exceed that size limit by 2 to 3 times for a single photo.
SMS/MMS is, in my opinion, antiquated technology. Apple had the right idea with iMessage, which seemlessly integrates with their phones' SMS/MMS interface yet allows for much larger files to be sent and can utilize wifi and faster mobile data technologies for message handling (traditional SMS/MMS is carried over voice channels, it doesn't even utilize 3G/4G data). Most iPhone users don't even understand that there is a difference between iMessage and SMS/MMS...which is why it works so well (for Apple owners). It would be great if there truly was a universal solution similar to iMessage yet available to all smartphone users...Google Hangouts may one day get there once it integrates with SMS/MMS as Google has claimed it soon will, but I'm not holding my breath on that one until I've seen it in action.
Regardless, if you want to share the awesome pics you take with your S4, MMS is probably not the route to take. Email does work much better.