Remember that the strength of a electric field is voltage over distance. That means decreasing the distance makes the field stronger as much as increasing the voltage. Also keep in mind that what you seem to be talking about is lethal voltages. These can seriously hurt or worse with just a few pF between them and ground.
In college one of my frieds had a neon sign transformer. That put out around 6 kV if I remember right, and was useful for all sorts of messing about with high voltage. We made a Jacob's ladder with it, and used it to make the anode voltage of a old TV tube we repurposed into sortof a oscilloscope.
Starting with a transformer is probably the best way. You get to high voltages quickly, and it's inherently isolated. The isolated part makes it a little more difficult to kill yourself. The output is also AC, so you can use a diode/capacitor ladder to step up the voltage easily.
To make a volume with your strong E field, put the high voltage on two opposite plates of some conductor, space them apart, and your high E field will be in the middle. Beware that there will be a force pulling the plates together proportional to the E field strength.