The difference between rounded and unrounded vowels is visually testable. Taking the examples feed [fid] / food [fud] as a basis, observe yourself in a mirror saying "feed" and notice how your lips are spread -- this is why photographers have you say "cheese", because it looks like a smile. That is a non-round vowel. Compare that to "food" where your lips protrude into a circle -- that is a round vowel. When you get to words like "cot" and "caught" this diagnostic may fall apart because lip position also correlates with vowel height.
For front/back, you can't generally see it, but maybe with a flashlight and a mirror you can convince yourself that your tongue is further forward with you produce the front vowel of "teeth" [tiθ] versus the back vowel of "tooth" [tuθ]. By comparing where your tongue seems to come close to the inside of your mouth in words like "fit, break, check" [fɪt brek tʃɛk] with front vowels versus "foot broke chalk" [fʊt brok tʃɔk] with back vowels again you can get some idea where "front" versus "back" vowels are. Likewise, comparing pairs like "seat" vs. "sit" [sit , sɪt] "late" vs. "let" [let, lɛt], you can get a feel for the difference between tense vs. lax vowels.
One warning is that this exercise will not work if you are not a native speaker of English though there are analogous exercises and examples for other languages. The other is that there are very many dialects of English, so perhaps the vowel that is phonetically [u] for me is [ʉ] for you. Because the actual pronunciation in various languages and their dialects is so variable, it is very hard to explain the phonetic properties of a vowel by citing a comparison in English (possibly the worst language as the standard for phonetic vowels, given dialect variation).
Instead, we learn these properties by comparison to a standard: the International Phonetic Alphabet. Fortunately, there are some expert pronunciations of standard vowels. Here is a page with all of the IPA chart plus pronunciations by Peter Ladefoged, and here is an analogous chart by John Esling.