If you search for 'transistor flip flop circuit' you will get lotsa schematics, mostly along the same theme.
**broken link removed**
These cover the basic idea, but you need a modification so the circuit always powers up in one state. This can be done with the right size capacitor replacing the left switch. Also, the left LED can be deleted and the resistor connected directly to Vcc like R1. I'll try to crank out a schematic specifically for your application.
I applaud your wanting to use transistors. Other options are a single CMOS flipflop chip like the CD4013, a two-gate flipflop circuit with a CD4093, a 555, and or course a microcontroller like a PIC or Arduino. But where's the fun?
ak