solid state has many more parts to make the RF generator
I'm not sure. whichever is cheaper I guess
The traditional way is to use beer bottle caps. You take the beer bottles and fill them with high salinity water, then place them in a tub of high salinity water. It's somewhat of a crapshoot, since a defect in the glass can cause a breakdown, but that's what safety cages are for. People often use a neon sign transformer (with a ballast, shorting them can destroy them) to charge them up.
*These capacitors are very dangerous due to high voltage, low inductance, and low ESR. They may not be very big, but they'll have a heck of a peak current. They can also shatter quite violently.
Edit: For a spark gap, I used two bottle caps and two screws. You can adjust the length of the gap as well as blow air through, and it's very cheap. They'll only last about 20 minutes of operation though.
If you search around google, you may find more specific plans. In addition to getting your circuit tuned, you also have issues with the best gauge of wire to use on you secondary, the size and shape of your topload, and all sorts of crazy things.