Your standard off the shelf AA batteries supply 1.5V each while rechargable (NiMh or NiCad) cells provide 1.2V each. So 2 or 3 of either type fits within the range.
You don't need a resistor as the circuit will only draw as much current as is needed to run it. It won't pull too much current unless you have a short. If you want to protect against that, or your circuit pulling too much current from the batteries at one time you can add an appropriate fuse in line with the positive voltage.
Looking at the BT schematic it looks like it's using a boost converter to step up the battery voltage, and I don't see a fuse, but I think the boost converter is current limited.
Easiest way to connect the board is to probably get a 2AA battery clip with wire leads and connect them to the contacts. There's a diode on the incoming line so I don't think you'll hurt anything if you accidentally put them in backwards, but check anyway