Looking for a transistor to power a load with PWM from a Bluetooth module
You are right that you need a transistor to switch lots of current from a GPIO pin.

It is often better to put the transistor at the negative side of your load.  It means that the GPIO pin does not have to see the 3.7 volts, or have to go that high to turn your load on and off. 

It looks like your research was around BJTs, (bipolar junciton transistors, i.e. PNP, NPN). If you are switching large currents with these you sometiems use 2 of them to get enough hFE (this is called a Darlington pair), although an hFE of a few hundred is very common.

Have you looked at FETs (field effect transitors)?  These have the advantage of not having a diode drop across them, so they don't get so hot and you don't lose so much power in the switch.

There are loads to choose from.  E.g. FDN335N from fairchild.