How to design a circuit that creates high electric field between two conducting plates?
Sorry to do this you you.  We're all ganging up on you telling you bad things.  I'm going to tell you a bunch of them, and get it over with.


An oscillator doesn't work that way.
A 9V battery is a poor choice for a power source.  If you need 1mA at 1500 volts, then the primary side of the transformer will draw like 170mA from the battery.  It won't do that for very long.
An oscillator won't directly drive a transformer, even if you built a proper oscillator.  Most oscillators are intended to deliver a signal, not power.
An oscillator running at 10MHz will be awful fast for particles you are trying to move, and then see the motion.
"Series or parallel" is the least of your worries.





Figure out how much field strength you really need.  Figure out how much force it takes to move your particles, then calculate the force exerted by the capacitor plates on the particles.  This ought to help you along the way.  Maybe you already know this because it is part of a known physics experiment.  If so, then you should also find notes somewhere on how to calculate the expected power requirements - or even a note that says "we used plates sized X, powersupply Y volts and Z amperes."
Now that you have the voltage, figure out how much current you need.  It should be in some way proportional to the area of the plates and the area of the particles.
Now you have some idea of how much power you need.  The voltage times the current gives you the power.
Since you will start with a lower voltage and step it up, the current at the low voltage will go up by the same proportion that the voltage does.  Say you need 1mA at 1000V, and start from 10V.  You'll have to provide 100mA at 10V to get the required 1mA at 1000V.
Select a frequency, or range of frequencies.  Do you want your particles to vibrate in place, or would you prefer to see them drift towards on of the plates?  For vibration you need AC, preferably at a low frequency.  I don't know about you, but I can't see things that vibrate faster than a few Hertz, maybe a couple of tens of Hertz.
In either case, you need an oscillator operating at an appropriate frequency, and an amplifier that can provide your calculated power to the transformer.
If you want to see the particles vibrate, then you are done.
If you want to see the particles drift, then you need to convert the high voltage AC to DC.  That's done with a diode, followed by a capacitor to filter it nice and smooth.  You can also use a higher frequency for this.  It makes the transformer smaller, and makes it easier to filter the DC smooth.


That's a lot of stuff.

If you want, you can buy the powersupply you need instead of building one.  That's probably safer for you, given your lack of experience.  They might look expensive, but they are certain to be cheaper than a funeral.

In any case, I suggest you find someone who knows how to work safely around high voltage and work with that person and learn how to be safe yourself.