To suggest is to propose or hint. If you suggest going out for ice cream, nobody's going to object. If you suggest that your sister should get the frozen yogurt instead, she might think you are suggesting something about her weight.
The verb suggest is also used when people speak about scientific research. Experimental results are often used to suggest whether a theory might be correct. For example, if a patient takes a new drug and starts feeling better, that will suggest the drug is working. Suggest can also mean to drop a hint. If you find a polite way to suggest that the pot roast is inedible, you don't come out and declare it directly. You wouldn't want to hurt Mom's feelings.