Having a low price budget, my search for my first ham radio to buy has led me towards Baofeng. A quick Google search on baofeng showed a lot of results that said that certain model baofeng handhelds are illegal in the US. Is that true?
They are legal to use, but only on the amateur bands. (Of course, you'll need to get a license first).
There was a lot of debate on whether they were legal, but the FCC finally stated that they were. Thus the older search results you found, such as this one.
Since I wrote this answer, there's been quite a bit of FCC enforcement (on importers, not amateurs), and as a result, most new Baofeng radios sold for amateur use in the US in the past few years have adequate filtering, and comply with the Part 97 standards. (They also don't accept being programmed for non-amateur frequencies). While there are still a lot of bad radios out there, the situation isn't quite as bleak as I originally stated, and it's no longer true that any given Baofeng is probably illegal to operate.
Original answer preserved below:
The ARRL runs a booth at Dayton Hamvention since 2012 where people can submit their HTs to be tested for spectral purity. Over the years 2016-2019, 100% of the Alinco, Icom, Kenwood, and Yaesu HTs they tested were compliant with the standards laid out in Section 97.307. Only 7.5% of the Baofeng HTs they tested were compliant, with 27% being "borderline" (violating the limits by less than 3dB) and the remaining two thirds being more than 3dB out of spec (Source: QST January 2020, pp. 60-61).
Amateur equipment isn't type-certified; the operator is responsible for the compliance of their equipment. With a fairly reliable source showing 92.5% of Baofeng HTs in the wild failing to meet the standards of Part 97, anyone without access to a calibrated spectral analyzer can reasonably assume that any given Baofeng is not legal to use on the amateur bands.
Since one aspect of amateur radio is "home brew" ... anything that transmits a "suitable signal" in an amateur band operated by a licensed amateur radio operator, is legal. The key part of the "suitable signal" is power level and spurious signal levels, and includes permitted modulations in the portion of the band wherein the transmission occurs.