I think the answer to this depends on what languages you're learning, and also possibly on how many languages you want to learn.
If you're a native Italian speaker, and you're trying to learn Spanish as your first foreign language, then it would be silly to learn PIE roots. Spanish and Italian are extremely similar. A huge number of words are cognates, deriving from Latin in both languages. Done. No reason to go back to PIE.
I'm a native English speaker, and when I first learned modern Greek, I found the vocabulary pretty challenging. It was the first language I'd learned that had so few obvious cognates with English. It definitely did help to recognize that πατέρας (pateras) was actually the same word as father. If I'd been unaware that there was any relationship between the languages, then I might have seen the phonetic similarity as merely a coincidence, and taken advantage of it as a mnemonic. Understanding that they come from the same IE root makes things a lot easier, because then I can recognize other cases where the same phonetic changes are at work, e.g., πόδι and foot.
This saves work. The genetic relationship between English and Greek is distant enough that when you connect words like these, you basically are going almost all the way back to PIE. (But I have a hard time imagining that it would help to learn all the reconstructed PIE pronunciations and the spelling system for the PIE roots as used in etymologies.)
But there is only a certain amount of mileage you're going to get out of this. I'm going to learn γυναίκα (yineka) = woman based on words like "gynecologist." For words like σκύλος (skilos) = dog, there just isn't going to be any cognate.
If you were going to learn a large number of IE languages, including exotic ones like Hindi and Persian, then I would expect that you would get at least some pay-off from systematizing your knowledge by connecting to PIE roots.