For your camera, you need AF-S lenses with the AF motor built into the lens rather than the body. Sigma's equivalent is HSM.
Full frame is a sensor size and refers to 35mm film which has an image that measures 36mm x 24mm. FF DSLR sensors share these dimensions. Your body is a Dx sensor which is also known as an APS-C size sensor and is considerably smaller than a full frame sensor. There are advantages to both sizes.
In terms of focal lengths of lenses, for your Dx body, a standard length is about 35mm. Anything shorter is usually regarded as a wide angle, with anything much shorter than 20mm being regarded as an ultra wide angle. Anything over the 35mm "standard" length is a telephoto. On FF, the standard length is usually regarded as being 50mm, with the same rules regarding shorter and longer lengths applying (but ultrawide being shorter than about 24mm).
Rather than jumping in with both feet and spending loads on lenses, use your standard zoom (usually an 18-55) and see which lengths you use most. Once you know how short or long you want to go, we can advise you better on specific lenses. An idea as to your budget helps too!
