Diagnostic radiology is a 4 year residency (after 1 year internship) which involves all diagnostic imaging modalities (including 6 months nuclera medicine). We read x-rays, cts, mris, ultrasounds, nuclear medicine. Most diagnostic radiologists will do some procedures as well (you don't need to be an IR to do CT/US/MRI guided biopsies, paracentesis, thoracentesis, or even radiofrequency or cryoablation).
All diagnostic radiologists rotate through interventional radiology for several months during residency. Some do an extra year of fellowship after residency focusing on interventional radiology. These people can also interpret all diagnostic modalities, but are more intensely trained in invasive procedures such as angiograms with or without peripheral angioplasty, nephrostomy tube placement, biliary tube placement/stenting/dilatation, G-tube placement, abcess drainage, vascular access (ports, PICCs, dialysis catheter placement, etc). In most private practices, IRs are expected to interpret images in addition to do procedures. In academia and in larger private practices, they will mostly do procedures.
Nuclear medicine residency is seperate residency that is 2 years after internship and only trains to interpret nuclear medicine studies (studies in which a radioisotope is injected into the patient). These include cardiac studies, PET, bone scans, HIDA scansm, V/Q scans, etc. Nuc med programs are trying to get some more training in CT since PET scans are often combined with CT scans to better localize abnormalities. The main problem with nucs is that radiologists are qualified to read all the same studies plus everything else and thus the jobs for just nucs are more limited.
