How close can I place two transformers?
The importance of the placement varies depending on what each one is being used for (power, audio, relative signal strengths, etc) For example, a potential disaster is placing a low level audio input transformer next to a high level output transformer. A similar issue could be the power transformer next to an audio input (or possibly output) transformer.
Placing two transformers passing the same signal/level near each other is not much of an issue though.   

Generally, to minimise coupling the cores should be placed at right angles to each other (flux lines at 90 degrees to each other), and as far away as is practical. You will see this done with input/output/power transformers in many audio amplifiers.
Different transformers have different flux leakage shapes, learning about this is useful. Toroidal are generally the best in this area. Shielding (e.g. Mu-metal) makes a difference also, though more expensive.
A good way to test for satisfactory placement is to apply a signal to one transformer (with a load on secondary) and measure any coupled signal in the other, whilst moving it around. You may need to use quite a high level "transmit" signal, and a simple amplifier at the "receiver" to make the signal easily measurable.  

For power transformers, remember there are also heat and vibration considerations, so solid mounting/damping and ventilation needs to be looked at. I agree with Tony that touching is  not a good idea, so if it can be avoided do so.