The most expensive live centers I own are probably about $100 (MT-3 size) and have had them for about 7 years. They are "medium duty" units capable of holding pieces up to about 350lbs as I recall. Heavy duty units are made to handle several hundred pounds more. A typical hobby lathe will never see anything that heavy and a light or medium duty live center will be fine.
The body diameter of a medium duty live center in MT-3 size is about 2 to 2.25". If the stock piece is very short, the body can get in the way of the tool post so, it's good to have a dead center around if you ever end-up in that pinch. An extended point live center is handy to have. When using a dead center at the tailstock, carbide tipped is preferred. It needs to be lubricated generously. Some folks use graphite. White lead was common but not available anymore (as far as I know). Keep an eye on the temperature. If the lube dries up, that tip can get really hot, really fast.
Dead center with carbide tips cost a little more than those without. The carbide tip is much more durable when used at the tailstock end to hold a piece. If you're spinning between centers, the dead center at the spindle end does not need to be carbide tipped because there's no friction there.
The price difference between the two is not all that much last time I checked.
Ray