I spent 13 years designing electronics of this exact nature: three phase induction motor reduced voltage soft starters and variable frequency AC drives. I spent the last few of those years as a VFD applications engineer helping customers select and configure this equipment for various loads and industries around the country as well.
You will not be able to build something that is cheap and safe. The voltages and currents involved are well beyond the safety margin of a hobbyiest, especially someone who is openly avoiding buying commercial units in order to save money. Don't do it!
While the theory behind AC motor control is very straightforward, the detail level work (heat sink sizing, snubbering, gate drive requirements, de-sat protection, motor overload calculations, bus capacitor protection, etc.) can be quite tricky to get down, especially with heavy duty cycling and regenerative power modes which a carnival ride will CERTAINLY be generating. I strongly caution you against trying to build something of this nature unless you have significant experience not only in microcontrollers and embedded systems design but also significant experience in power electronics and three phase circuitry. People get hurt and killed building this stuff.
My first question for you is whether speed control is really required, or if you only require a soft start up and slow down. Do you vary the speed of the motor once it is started? If not, you may be able to get away with a MUCH cheaper reduced voltage soft starter. These units act like three phase light dimmers; they only adjust the applied voltage to the motor. You will not have a lot of torque at low speeds, but with the right design of motor (NEMA class D) you can achieve exactly what you're after with a fraction of the cost and maintenance.
If you really do need to vary the full-load speed of the motor then you are more or less stuck using a variable frequency drive. As you are aware these are expensive and if you buy cheap you are likely to replace them sooner due to your high surge current (they call this "constant torque") application. What I would definitely recommend doing if this is the case would be to contact various manufacturers (Allen-Bradley, Cutler-Hammer, SAF drives, Benshaw, Yaskawa, etc.) and ask for reconditioned units. Ask for a drive capable of delivering 150% rated current for 30s (this is usually known as heavy duty) or size the drive 30-50% larger than your nominal current rating. You will also likely be running off of generator power which is notorious for being undersized and prone to brownouts and surges as the load requirements change with the state of the equipment being run. Drives don't like that (voltage sags cause current spikes as the motor starts slipping and surges can cause you to overvoltage the bus capacitors) and have a tendency to either fault out or blow up.
I am all about the little guy building something and saving a buck, but this is not the type of project to do this on. If you really want to build a three phase AC drive, start with a little 10HP 480V motor with a hand brake on a test bench. You have all the potential for experiencing the pants-filling sensation of an H-bridge failure or a bus capacitor explosion two feet from your head but without the potential lawsuits and loss of life (except perhaps your own).
While the answers above are excellent, you have failed to identify the types of AC motors you are using.
Are they brushed motors? Or squirrel cage types or?
If your motors do not have brushes, then you should not realistically be looking to control speed, as they will naturally try to assume the speed of the windings/armature ratio versus frequency of the supply, and under a real load, you could end up cooking the windings and the triac.
If your motors have brushes, (ie, like vacuum cleaner motors, blenders, drills and saws use brushed motors), then it's a lot easier, since you're dealing with a commutator that chops the voltage to something like pulsed dc.
Best Answer
Voice of experience: Use a jackshaft and additional belts/pulleys, &/or change to a different motor (DC with DC speed controller or 3-phase + VFD (variable frequency drive) variable speed.) The DC motor and speed control can often be salvaged from a treadmill that someone gave up on using, for free.
The fundamental limitation of running induction motors slowly on household (60 cycle per second, 3600 cycles per minute) AC is that the motor type you have has 4 poles if it runs 1800 RPM - so you'd need 36 poles to turn 200 RPM on 60-cycle AC. That would be a very rare bird indeed. You're actually fortunate if your drill press motor is not 3600 RPM to start with...
The other limitations (which apply to "universal" motors that can more easily be speed controlled, but are terribly noisy by comparison)(and also to DC, and to a slightly smaller extent 3-phase + VFD) are terrible (worse than a linear reduction) torque, and poor cooling/overheating since the motor's (built-in) fans are not running at a reasonable speed to cool it.
You might find gearmotors (normal motor speed of 3600 or 1800 RPM and an attached reduction gear) that run that slowly, but you won't like the price, especially if you want much power/torque.
The initial cost of the drillpress you are starting from has little impact on the cost of doing non-standard things with it (and may make a more expensive model that has better features such as dual reduction or a wider reduction range already built in look less expensive in the end.) Then again, you may be starting with completely the wrong tool - metalworking lathes are not too hard to find used affordably in moderate sizes, and typically have a "back-gear" setup standard that offers very low speed and high torque. Good for coil-winding (at a guess since you don't say and this is EE.)