The best inexpensive "non-ham" radio is actually a ham radio, Baofeng UV5R (or BF-F8HP).
Inherently those radios are poor quality when compared to ICOM, Yaesu, or Kenwood handhelds, but they are dirt cheap (I saw Black Friday prices of $35).
Older versions of those radios could receive and transmit across the VHF/UHF bands, but about 7 years ago the FCC forced Baofeng to lock out the non-ham segments (e.g., FRS, GMRS, aviation, marine, commercial, military). I understand that it is possible to "jailbreak" the newer models, but we have the original models without the lockout, so I never had to jailbreak our Baofengs.
It's easy to jailbreak most ham radios by removing the lockout diodes. As with most jobs, it's a cinch using the right tools. There are online videos and .pdf files explaining which diodes to remove. If you remove or damage adjacent diodes or the motherboard, you will lose functionality or even have a dead radio, so be careful.
Caveat:
No license is required for the FRS bands.
A station license is required for GMRS, but no operator-specific license is necessary.
You are required to operate within the legal power limitations for the band.
It is illegal to operate outside of the allowed bands.
Many licensed hams are as annoying as Homeowner Associations and will "nark" violators to the FCC.
There are other options. If you had a different radio in mind let me know.