ESP -> connected to.
Vcc -> 3.3V supply (minimum 500mA & not much more than 1000mA as many supplies don't react fast enough to change when transmitting).
Gnd -> Gnd.
CH_PD -> Vcc or Vcc via 10K pull up.
REST -> Vcc or Vcc via 10K pull up or Vcc via 10K & 1K to GPIO16.
GPIO 0 -> Vcc via 10K pull up (can be shorted to ground to flash - ensure pin is not set as output and high when shorting or it will damage the output).
GPIO 2 -> Vcc via 10K pull up.
GPIO 15 -> Gnd via 10K pull down.
Tx -> Rx.
Rx -> Tx 3.3v - not all devices are 3.3V even if they have a switch or jumper.
I'm sorry if I'm stating the obvious and you have addressed all these issues but with the amount of infomation on the net (not all correct or ideal) and the lack of a schematic, I thought I'd check.