There are no specific precautions required for staying healthy in Europe as most restaurants maintain high standards of hygiene and in the majority of countries tap water is safe to drink. However, for more precise details on these matters as well as for general information on emergency care, pharmaceutical regulations and dentistry standards etc., please consult the 'Stay safe' section on specific country articles.
EU/EEA citizens should apply for (or bring) the free European Health Insurance Card (EHIC), which grants you access to state-provided healthcare within the European Union as well as Norway, Switzerland and Lichtenstein either at reduced cost or free of charge, under the same terms as a resident of the country you are visiting. If you are used to free healthcare in your own country, remember that some member states expect patients to pay towards their treatment, and you may be expected to do the same. And do remember that the EHIC does not equal a travel insurance; it doesn't cover private healthcare, the cost of mountain rescues or repatriation to your home country. Neither does it allow you to go abroad specifically to receive medical care.
If you are not an EU/EEA citizen, remember to buy a travel insurance policy, while some countries do provide free emergency care for visitors, any follow-up treatment and repatriation is your own responsibility, and some countries expect you to foot the entire bill for any treatment yourself – the fabled universal healthcare system does not equal free treatment for non EU citizens.