Can I transfer my iPad 2 3G sim to an iPad mini?

An iPad 2 takes a micro-sim, iPad Minis take smaller nano-sim cards : iPhone and iPad: Understanding which SIM card your iPhone or iPad uses

You might be able to cut it to the correct size, but you might be better off asking your carrier for nano-sim.

Not directly, as they use different sized sim cards. However your carrier should be happy to transfer your current service to a compatible Nano sim card for the mini at little or no extra charge.