Coinbase or Gemini would be the most trusted options in the US at the moment, but both of these are custodial exchanges so if keeping full control of the money in your account is important, these would not be ideal.

For MoneyGram/WesternUnion exchanges, LocalBitcoins is by far the most popular. They have now introduced heavier KYC as well, so should hopefully be a little more reliable than before.

Honestly I'd suggest use LBC but just look for people with high trust scores and many reviews. Start with small transactions, and never risk everything with one trader.

Edit: When it comes to fiat exchanges, you can basically have either complete safety, or complete control. No exchange is going to provide you safety (like Coinbase with insured deposits) without taking some control away from you. So the choice is yours as to which of these two options is most important.