I just purchased a elliptical machine and they recommend for it to be on a 15-amp grounded outlet with only the elliptical plugged into the circuit. I am installing a dedicated 15-amp circuit and my question is this, should I run a 12 gauge or 14 gauge wire?

If 15 amp is recommended, then #14 is fine, however #12 does not cost than much more.

Just plug it in an existing circuit. If you have conduit, and have the ability and easy access to the receptacle, go for it. But in 99% of all installations, you just need to plug it in wherever you want. After speaking to a Nordic Trac installer, you may want to get a surge protector, if the equipment is electronic, which I am sure it is. The other advice the installer gave me is to unplug the unit whenever it's not used. I'm sure that is overkill, but a lightning strike will no doubt affect the neutral and the ground as well.
(But I never unplug my computer)