Picked up my turbo last night and used it without charging.
First mistake.
I plugged it up to the factory charger at 5%.
Second mistake.
This morning I took it off at 100% and have checked email, played on Facebook and shopped online for about an hour. I'm already down to 87%. Does this battery get better with time?
No, but the phone stops its initial settling after a few days. It has to sinc everything on the phone while you're doing your stuff, so that takes a lot of battery. Typically the first percent should take anywhere from 15 minutes to half an hour (IOW, you'll be down to 99% before the first hour off the charger is up).
Motorola has
all this online. The "charge it before first use" part. (This is true with any rechargeable battery in any device - when you get it, or when you get a new battery, always charge it before using it. I won't even let them set a phone up for me with the battery I'm taking home. I cvan set it up myself after the battery is charged or, if they have to set it up in the store, either use another battery or take the battery out and set it up running on the charger. It's one of the reasons my 10 year old spare phone is still using the original batteries [I always carry a spare, and won't buy a phone I can't swap batteries in when I'm out in the street.])
Now let's see if we can correct the damage your mistakes caused.
Run the phone until it tells to charge it. Turn it off and let it fully charge. Use it until it tells you to charge it again. (It doesn't have to stay on. You can turn it on and off as you like, but don't charge it again until it tells you to.) Do 3 full cycles like that fully charge, use until it tells you to recharge.
After the 3rd charge, try to
never let it get below 40% charge. The longest life (and with a non-removable battery that's especially important) will be if the battery always gets charged around the 50% point. If you keep draining it down to 5%, you'll need a new battery by summer. I haven't seen any prices for that battery yet, but I'm guessing it will be around the $50 range, and another $50 to open the phone and replace the dead battery - if it's a local repair shop and they want to start getting your business. That's every 6 months, if you keep draining the last bit of power out of the battery. For a battery that should last at least a few years if it's properly taken care of.
If you need the battery to keep working longer than it takes to get down to 40%, you bought the wrong phone (one you can't swap a spare battery into), but the battery doesn't care that you need 95% of the power between charges, you'll still kill it fast.
While we're on the subject of batteries and Droid Turbos, it uses Qi wireless charging, so you can use any Qi charging pad - it doesn't have to be one especially made for the Turbo. If it has a red color scheme and Verizon's or Motorola's name on it, you're probably going to pay a lot more - but it's the same thing inside. $20 for a charging pad is top dollar today. (The $13 Lugu Lake [I've been using one for months] is just as good, and the $7 GMYLE, although it's smaller, so it'll take a bit of practice learning just where to sit the phone, will probably work just as well.) No reason to pay $45 for a charging pad, unless you like a vertical easel and pad in one, and you can make one out of a little $5 plastic nick-knack stand (or $1 worth of plexiglass) and a cheap charger. Or two - one for portrait, one for landscape. Just cut a hole in the plastic (so the face of the charger is flush with the
front of the stand when it's in place), glue on 3 little "ears" to slide the charger into, and for $8 you have a $45 charging stand.