"find" means search but "find out" means understand and realize "check" means control something but "checkout" is the time when you want leave a hotel or you finished your buying in a shop and you want pay money.

"Find out" and "check out" are phrasal verbs which have similar meanings to "find" and "check". I would say the phrasal verbs are a little more specific, where "find" and "check" are used in a more general way. For example: 1. I am going to find a pair of shoes. 2. I am going to find out how much a pair of shoes cost. 3. I will check if I can find a doctor. 4. I am going to check out the doctor's experience.

“When will the next train leave?” “Let me check” (if easier) “Let me find out” (if harder) “I want to check out the food here.” (Here’s a place that has food and I want to try it to learn whether I like it) “I want to find some food” (I don’t know where I’ll get it. I have to look)