If I level up my eidolon's ride skill, can I have it ride into battle?
Assuming you're talking about a bipedal eidolon, and not a "humanoid" eidolon:
According to the PFSRD any points in ride describe a creature that is "skilled at riding mounts, usually a horse, but possibly something more exotic, like a griffon or pegasus. If you attempt to ride a creature that is ill suited as a mount, you take a –5 penalty on your Ride checks."
It stands to reason that any creature with a ride skill would be able to ride a mount. The logistics of a limbless being, (for example, an aquatic eidolon base form) would be worse at it, but the exact nature of it seems to be up to the DM.
I would rule as a house rule that creatures without arms would have a -5 to all ride checks, minimum, but I can't find any evidence of something like this being official.