See especially Eating Meat of People of the Book (ie supermarket meat does not qualify):
a. The one slaughtering be a Muslim, Jew, or a Christian.
b. That the name of God be consciously uttered when the slaughter takes place.
c. That the animal be slaughtered at its throat.
d. That atleast three of the four main veins of the throat be cut with a sharp object, such as a knife.
If one can ascertain that the above conditions are being met, it would be permitted to eat such meat. If unable to, then one should avoid and veer on the side of caution. The ruling given in our times is that kosher meat is generally permissible to eat whereas Christian meat is not.
One thing to recall is that eating meat for most of us is a luxury and not a necessity and the health benefits of avoiding its excessive consumption are well known. The Prophet (Allah bless him and grant him peace) and his Companions (Allah be well pleased with them) were known to eat meat rarely and scarcely.
And Being Vegetarian:
If it is done with a good intention then there is no harm in it. So if one stops eating meat because the animals are not treated well before being slaughtered or the like then that will be fine.