As you say, Monaco is part of Schengen Area, which means there are no internal border controls between Monaco and its only neighbour (France). So you certainly do not need a visa. No one's even going to look at your passport. Quoting Wikipedia:
Monaco has an open border with France. Schengen laws are administered
as if it were a part of France [...]
I'm not sure about EHIC. But you have some sort of travel insurance from your home country, don't you? If you do, it very likely is valid in Monaco too. If you don't (and you travel), consider getting one.
Edit: UK's Foreign & Commonwealth Office has a Monaco info page that directly answers that part of your question (emphasis mine):
Monaco is not a Member of the European Union and therefore EU
reciprocal medical arrangements do not apply. The European Health
Insurance Card (EHIC) does not provide health cover in Monaco. It is
essential that you have comprehensive travel and medical insurance
So do get a travel insurance. Other than that, there is nothing about going to Monaco you should especially worry about, even if you'll technically be outside EU.
Regarding your final question about getting to Monaco: it is easy and there are lots of options, as e.g. Wikitravel will tell you. For instance, you could take a train from Nice, Cannes, Menton (France) or Ventimiglia (Italy). Or take bus 100 from Nice, which I can recommend: it takes a lovely seaside route and costs just 1 €.
(If still in doubt, you should ask more specific questions about getting to Monaco from a given place!)