Creating positive and negative voltages from a battery
There are off the shelf chips that do what you want.  Also, you don't need any transformers since there is no need for isolation in this case.

For the positive supply, you need a boost converter.  This is assuming you connect the negative side of your 3.7 V battery to ground.  There are also switcher chips that are intended for making a negative supply from a positive one.  If your negative current demand is low enough, a charge pump might be all you need.

These things are available off the shelf.  For your low voltages and currents, you can find chips that contain the switching element.  Generally you supply the inductor, a few caps, and sometimes a external diode or two.  Look around at offerings from Microchip, TI, LT, Linear, and many others.  Or use a distributor web site to drill down parametrically across the vendors carried by that distributor.