I had an idea of making electric current from bacteria. There are some bacteria that can photosynthesize, and some others are capable of transmitting electrons through each other. Is it possible that through genetic engineering, we could insert the genome that allows bacteria to perform photosynthesis, with other capable of transmitting electricity, and therefor making a bacterial culture, big enough to make a free energy source?
What you're describing is a phototrophic biofilm microbial fuel cell! They're pretty neat.
Essentially the sun powers the bacteria and the bacteria in the anaerobic side passes electrons through the graphite electrode to use oxygen at the cathode. There's another explanation here. You could also "unfold" cyanobacteria and layer them and you'd probably get an efficiency increase but the system would be pretty fragile. Compared to other MFCs, though, everything is fragile. Dirt even works(not very well, but it's dirt).
You do not need to merge anything. Photosynthesis already includes the electron transport across the inner membrane of the chloroplast (in case of plants) or across the bacterial membrane (in case of bacteria), creating voltage difference between inner and outer side. The only problem, this voltage is relatively difficult to take away.