Winkie131. First thing to do is to go into your "profile" and update it, putting your city/state and USDA zone. That way, every time you post, we'll know where you are. Location is important. I know you said that you were in GA, but unless someone goes back and reads this first post, they'll never know.
If you are going to cut the tree back, you need to do so now. In 2-3 months, our plants will be headed into dormancy, so you want the rooted part to have time to put on some new growth (tips) and leaves. If you can't do the cutting now, it would be best to wait until late spring/early summer. If you do take that cutting now, you won't need to water that "stub" much at all and I would definitely keep it shaded between 11:00 AM and 4:00 PM. The stem will likely sunburn without those leaves atop it. I would say water it when the top couple of inches of soil is dry. To let it go completely dry will induce dormancy.
I can't tell for sure, but is all the "wood" of those three tips, green? If so, not having any mature wood (which would be more firm and grayish) makes rooting the individual tips difficult. I have rooted green-wood, but it can take quite a long time, time you really don't have. Thus you'll want to root the single, large stem with the three tips on top. I don't know why, after three years, those tips have not grown more, with more "woody" stems. Do you know what variety you have? I don't know any way to really make a plumeria bloom. They have their cycle but a lot depends on their growing conditions.
There are lots of ways to root a cutting and I don't know that there is a "best" way. For years, I rooted in 100% coarse perlite, with not a drop of moisture, other than to spritz the stem/tiny terminal leaves with clear water every few days. I have now started bagging my cuttings and I am primarily using damp, long-fibered sphagnum moss in that baggy. It remains to be seen whether bagging produces a rooted cutting any faster than the perlite way. I will know more by the fall.
Books have been written about growing plumeria, and there is no way to cover more than just cursory items here. If you have specific questions (depending on what you do right now), those are more easily answered.
