It sounds like more than one problem. Yes, the ignition is separate from the battery, but the engine must be turning over at a decent speed for the ignition system to work, and that has to be done by the starter. So if the battery is dead, it won't start. But, you said (if I understood correctly) that you charged up the battery, and the lights and everything worked, except the starter. That would seem to indicate the battery was not dead, yet the starter would not turn the engine over. Now, on a Rebel, when you turn the switch to on, the lights come on. But when you push the starter button, the headlight shuts off until you release the button. That is so all available battery power can be used for the starter.
So when you push the starter button, does the headlight turn off? If not, you have a problem in the switch. I suggest jumping the starter relay, what some people call a solenoid, with a heavy piece of cable. The ignition switch should be off, the battery should be fully charged, and the lights should work (with the switch on, turn off the switch once you have confirmed the lights work) If the starter turns over when you jump the relay, then the starter and cable from the starter to the relay are good.
There are 2 smaller wires that operate the relay (a relay is an electrically operated switch. It is necessary because there is no way the small pushbutton starter switch can handle the current required by the starter) Using a multimeter, see if you are getting 12V across the 2 smaller wires when you push the starter switch with the ignition switch on. If you do, then you have a bad relay. If you don't, then the problem lies somewhere between the starter button and relay. Check the fuse first. If that is good, then you will have to backtrack to find the open circuit.
While the Rebel has a very simple electrical system, there are a lot of things that can be wrong, and a lot of places current has to go through before it reaches the starter. First of all, you have to get current from the ignition switch to the starter button. Then it has to go through the kill switch, clutch nanny switch, and sidestand nanny switch. There can be an open circuit in any one of these places. But I would make sure the light goes out when you push the starter button with the ignition switch on first. If it doesn't, that indicates a problem with the starter button.
Also, your new battery seems to have gone dead awfully fast. You may also have a problem with the charging system. Do not confuse the charging system with the ignition system, they are completely separate. But if you are riding with the lights on, and the charging system is not working properly, the lights will quickly drain the battery.
A must have tool for checking electrical circuits is a multimeter. It does not have to be an expensive one. You can get them from $10 to the skys the limit. The $10 one will work fine. I prefer the analog type, as the digital type can give you erroneous readings. You can get a small reading when it should show zero, just from bleedover voltage. That can make you think you have something when you really don't.
I have several multimeters, but this is the one I grab most of the time Power Gear 500-Volt Analog Multimeter-50952 - The Home Depot