I agree, I've got a Baofeng UV-5R and while it's not a bad radio overall for the price, it's got the dreaded harmonic issue too. It sounds excellent on the band you're actually trying to transmit on, but I only use it as a backup since I don't want to cause issues on other bands.

I'd say to the prospective new ham, spend just a few more bucks and get an Anytone or Tera radio. They are well made, full featured, and still half the cost of your average Yaesu or Icom radio. Barring that, look for well-kept used gear on eBay or at hamfests, or borrow a good radio from a local ham. There aren't many of us left, but I'd still say everyone knows someone who knows a ham.