After tinkering around with my phone, the Photos app, iPhoto on my computer, iCloud settings, and Photo Stream, here is what I've been able to come up with.
I took all of my photos from my iPhone and uploaded them to iPhoto on my MacBook Pro.
I then created a new Photo Stream (I just called it "All Photos). I only added myself on the user list of my Photo Stream.
I selected all of my uploaded photos, hit "Share", and uploaded them all to iCloud. I then added them all to my Photo Stream.
I went into the Photos app on my phone, tapped Shared (bottom, middle), and there was my Photo Stream that I created, with all of the photos I had uploaded.
I then deleted all of my locally stored photos on my iPhone. I started off by just deleting a couple then checking on my Photo Stream to make sure they were still stored and viewable.
Now, whenever I take a photo with my iPhone, it stores locally. It only goes into my Photo Stream if I open the Photos app, tap "Photos" (bottom, left), navigate to the photo, then tap "Share".
If I delete a photo out of my Photo Stream, if I had that photo stored locally on the phone as well, it only deletes out of the Photo Stream - it does not delete the local copy.