Alpha radiation is large -- helium nuclei, and can be blocked by aluminum foil. It is ionizing nonetheless and if very strong will harm you. Beta radiation is high energy electrons (the stuff that makes your CRT and TV light up) and it also can harm you as ionizing radiation if strong enough. Gamma radiation, like x-rays is the highest energy and most damaging (hardest to shield-- thick concrete, lead, etc., etc.)
Bottom line is that there must be something that is radioactive (i.e. fallout) to emit the radiation. If you ingest radioactive material, you are at risk. If you touch it, or it touches you, or just gets near you (depending on its emissions and strength,) you are, of course, at less risk. "Radiation" does not hang around. It has to have a source. As I understand it, the nature of a neutron bomb's fission is that is does not cause other atoms to become radioactive, but rather just produces an intense stream of neutrons upon its explosion, thus destroying the ability of biological nuclear material (RNA and DNA) to sustain life. IT also produces significant EMP, shutting down unshielded elctronics.