食品, 食料品 and 食料 (食糧) all potentially have same referents, but the circumstances where you see those words are significantly different. If I restate their concepts in clearer phrases:
- 食品: items for you to eat
- 食料品: items for you to buy and eat
- 食料: resources to sustain lives
食品 is probably the word that covers broadest meaning to generally refer to foodstuffs, as when you abstractly imagine or discuss "food", this is what you mean. But in everyday situations, it mostly means final, or near-final products that are ingestible with at the most a processing you can do at your home's kitchen. Food used as an ingredient of cooked meal is called 食材, in particular.
食料品 is best translated as grocery. Foods on the store shelves. Though it's a hyponym of 食品, we use this one more in this specific sense. Imaginably, we rarely say 食料品を食べる, because, you know, it makes me think of such kind of act.
食料 or 食糧† is from a more survivalistic perspective. Food as material to keep you alive or working, as if fuel for human. In the situation you say 食料, categories of food are usually irrelevant; only the amount matters. When we say 食品問題, it may mean food safety or food waste and such things, but 食料問題 almost solely refers to concerns of food supply capacity. For this reason, the translation of "food" you see in games such as Fallout or Age of Empires is always 食料.
By the way, we have a native word 食べ物, which can cover all of their usages.
† It may need a little explanation. 食料 and 食糧 were originally different words, that had separate entries in pre-WWII dictionaries with different kana spellings, しよくれう and しよくりやう. However, as the result of kanji restriction, 食糧 has been officially to be respelled as 食料, and largely taken over the word form. What I describe in this paragraph is what 食糧 used to mean. Nowadays, kanji restriction is more and more loosened, so that the spelling 食糧 is also used at large, but 食料 still remains an alternative spelling of 食糧, as its original usage has mostly migrated to 食料品.