As the others have said, going publicly is pretty much free, except for the GP visits, and prescrition medication.
When in hospital, you're charged €75 per night, up to a maximum of 10 nights per year. This covers all costs involved in your treatment including the doctors, nurses, surgery, accommodation, food, etc. Medical card holders are exempt from the €75 per night charge. The Citizens Information website has info on public hospital charges [broken link removed].
The main bottleneck in the system at the moment is getting to see a consultant, as well as out-patient scans and tests. You can choose to see a consultant privately and pay for it, but your placement on the public hospital waiting list will be the same place you would have been had you seen the consultant on the public system. (I checked this with the Dept of Health and this is their official response - .) However, truthseeker's experiences seem to be different, so doctors probably take that as guidance rather than a hard and fast rule and make their own judgement calls.
The National Treamtment Purchase Fund is definitely worth considering once you've had your consultant's appointment. They look at the cases of people waiting more than 3 months for treatment. Even if they can't help you, it's no harm contacting them anyway. The website is
www.ntpf.ie.
To be treated as a public patient, there's actually nothing you need to do. Once you've gone to your GP, s/he will refer you to a consultant as appropriate and from there you're in the public system.
Finally, if you're a PAYE worker, make sure to claim tax relief on the charges (GP, hospital, medication as well as consultants should you go privately).