Small scale sugar cane syrup production was common at the turn of the nineteenth century. If you're interested in harvesting and pressing your cane today, you'll need to find or build a press.
Mechanical presses work much the same way a fruit press or even a clothes wringer or pasta machine would. You need to create pressure using gears to pull the canes through grooved drums or, alternatively and less effectively, use a lever to press the cane. The dry stalks are kicked out and sometimes used as fuel. The juice produced from the crushing is drained into a tub, collected and boiled down into a syrup. Early mills or presses sometimes used mules to turn the wheel to press the cane. The best way to construct your own is to carefully study, in person if possible, models in existence. You'll need a working knowledge of gear design. Metal mills will need outside tooling; wooden ones may be constructed by a skilled workman.
Small mills can also be run by motor. Short of developing your own foundry, your best bet for making your own powered sugar cane press is to find and refurbish an antique. Look to Florida, the Caribbean Islands and the Philippines for proper cane presses. You might find cane mills or sorghum mills on old farms in American Southern states, like Georgia. Search for names like "Golden's" or "Cook." These companies made and sold presses from the mid-19th to the mid-20th centuries. You can convert horse-powered mills to belt-driven mills with kits or some salvage and lathing work.
Several organizations have grown to meet the needs of people interested in making syrup or molasses from sugar cane or sorghum. Living history farms in the South sometimes have demonstrations of the pressing and boiling process. Check with groups like Syrupmakers for information on getting into production.