Can I use any FX lens with Nikon D5100 ?

Based on reading only it seems like FX lenses cover better range as compared to DX lenses.

Since D5100 is a DX camera, can I use any FX lens with it ? If yes, which one is a good lens for a beginner ?

You can use them but you will not have Auto Focus. Your D5100 will only be using the central portion of the lens and not the full lens.

As long as the lens is AF-S, it should auto focus just fine, shouldn't it?

Yep - what they said. A good FX lens can be a lifetime investment, and I'm glad that my DX-only glass purchases were very limited when I started out.

. All the AF-S lenses will auto focus and meter with the D5100.
. All the AF & AF-D lenses will meter but not auto focus with D5100.
. All AI and AIS lense will work but with manual focus and manual metering.

I have all the three types and find that the kit 18-55 VR-II is the least demanding lens for the 24MP sensor. No AF and no VR means that in most cases to get razor sharp images I have to
. Use high shutter speeds to reduce the blur due to wind
. Mount the camera on a tripod to reduce blur due to had or mirror.

If you have steady hands and the subject is not moving much then sharp photos are possible, even at low speeds, but if you wear glasses with high power (out side the diopter adjustment range) and your hand shakes a bit, then you are stuck to high speed and/or flash for hand held shots, if you have no AF or VR.

Thank you guys for your comments - all the comments are quite helpful.