I have Seagate Expansion external hard drive and Lenovo Tab 3 10. Can I connect the hard drive and watch movies and photos on it?

No, not directly. There are no USB or similar ports on it for doing any external drive connection. This has to be a guess since nothing in the "Tech Specs" describe what I can see in pictures that might be USB or HDMI or some other ports. Could you? Not sure.
But you might try wireless network from a computer that has that external drive attached.

You need an OTG cable that will plug into the microUSB of Lenovo and the USB of the external drive. Alternatively if it will take a micro SD card copy movies and photos onto the SD card and view them that way

Buying the external cable is not a problem. Just to be sure it will work out with it.