Getting to Elba is easiest from the main departure port of Piombino, just south of Livorno. You can arrive by train car and bus fairly easily, and once there you’ll find no shortage of offices selling tickets for the ferries, or traghetti, departing for Portoferraio Elba.
Ferries are numerous being as often as 30 mins (in season) and every 2 hours
out of season. The cheaper option seems to be Toremar with the other company Moby ferries being the more common option (you get a sticker slapped on your car, plus they paint their ferries with bright colorful shapes, which I quite like).