So, am I correct to say that, to limit the current to 1.26A (Rating of the battery I have) the LM317's Output goes through about a 10Ohm resistor, back to the Adjust pin, then that output (Which I hope would be limited to 1.26A) is run through a second LM317, which is voltage limited to 13.5v?
Yes, exactly.
The first will make sure current isn't above 1.26A, and the second will make sure voltage isn't above 13.5V. At first, you should see the voltage rise slowly (as 1.26A is delivered), but when it's approaching 13.5V, this slows down, as the second regulator is starting to limit.
Note that it should be a 1 Ohm resistor, not a 10 Ohm. There are calculators you can use to check stuff:
http://www.reuk.co.uk/LM317-Current-Calculator.htmAlso, note that the resistor needs to be able to handle the current that flows through it, the calculator will show you which rating you need.
And finally, for needing a 1 Ohm resistor capable of 1.57 Watt, you can instead use 2 x 2 Ohm resistors, capable of 1 Watt each.
I'd probably go for some safety-margins, and either get a bigger resistor, or multiple smaller ones. If you need to be able to handle 1.57 Watts, might as well aim for 4-5 Watt rating, such as 4 x 4 Ohm resistors.
There's also huge things, which should keep you well covered:
http://www.ebay.com/itm/2x-1-OHM-50W-Wirewound-Aluminum-Housed-Resistor-50-Watts-/400310239042?pt=Vintage_Electronics_R2&hash=item5d34597f42Oh, and finally, with this, you'd have a voltage-drop across both regulators, and the resistor, so your input-voltage needs to be high enough to allow for that.
Sorry, still learning. 
Learning is good.

tld
CAUTION: Keep in mind that lead acid batteries can supply insane amounts of current, so be safe at all times. I recommend permanently attaching a fuse, and only accessing the battery through the fuse, for either charging or discharging.