High Voltage (3,000V) Capacitor Charging Circuit Needed
Charging a 200 uF capacitor to 3000 volts in 5 seconds allows you to estimate the current needed using this formula: -

$$Q=CV$$  therefore

$$\dfrac{dQ}{dt} = C\dfrac{dV}{dt}$$ And \$\dfrac{dQ}{dt}\$ equals current

So, plugging in values for C, dV and dt we get a charging current of 120 mA.

This is what your secondary voltage has to supply but, given that charging will tend to reduce as voltage gets higher, you should consider increasing the current by maybe 30% and having a circuit that could produce an open circuit voltage of maybe 4000 volts.

This leads me to estimate the output power as something like 500 watts.

So, you need to go to some site like Ferroxcube's and estimate the size of the ferrite core you will need based on power and switching frequency. The site can guide you to a suitable ferrite core material and size and then you'll need to find a particular core made in this material and having the correct overall size for delivering this power.


  I would also like to know what the approx. weight of the transformer
  might be


Investigation of Ferroxcube's producs and materials will give you weight but be under no-illusion - it will take you hours of head-scratching and the learning process can be steep.


  if I wanted to charge multiple capacitors of the same size in
  parallel, would I need multiple separate charging circuits


It depends on step 1 - finding a suitable core.