Listening on amateur radio frequency bands is legal, but transmitting requires passing an exam and being licensed by the Federal Communications Commission (“FCC”). If you're never had an amateur radio license before, there are three levels of exams that can be taken:
The exams build on each other, so you must pass Technician before General, and pass General before Amateur Extra. If you pass an exam, many examiners will allow you to take the next level during the same testing session for free, so you may want to be prepared to take the next exam.
The easiest exam to pass, this allows full privileges on all line-of-sight bands with only a minor exception (repeating digital traffic). It also allows full privileges on 6m and a small bit of SSB phone (a form of voice), and a smattering of CW (Morse code) privileges on some HF bands.
The 6m band can go thousands of miles if the conditions are right, but it's quite unreliable. So if you're just intending on talking locally or on repeaters, the Technician exam will provide you with that.
This is the first license that truly opens up your ability to use the HF bands and talk worldwide. There are very few restrictions, other than not being able to transmit in ranges designated only for Amateur Extra.
With how similar the material is for Technician and General class, and given how much more you can do with the General class license, it's often a good idea to study for both exams and pass both at the same time.
The Amateur Extra license class gains two privileges. First, you gain full access to the entirety of the frequency bands with no restrictions. For contesters or Field Day (ARRL awards points, but says it's not a contest), this extra bit of the bands is crucial. But for ham radio operators that don't participate in contests, the only benefit is that you don't have to remember where the edges of your band privileges stop.
That said, many countries have reciprocal agreements with each other, allowing amateur radio operators to transmit in each other's country. For many countries, you must have an Amateur Extra license to operate at all in that country.
If you just want to listen and not transmit, or even to own amateur radio equipment, there is no requirement to be licensed.
If you transmit and aren't licensed, amateur radio operators will ask for your callsign, and when you don't have one they'll ignore you. A ham operator can lose their license, and potentially be fined, for speaking with unlicensed people on ham bands.
But that said, there are plenty of people who buy a $30 handheld radio to talk to their friends illegally and never get caught. No different than when CB operators modify their equipment to use huge amounts of power… The FCC isn't going to come knocking on their door.
Where people start to get into trouble is when, though their lack of knowledge and skill, accidentally start interfering with licensed use or interfering with health and safety communications. The FCC does track down those people, seize their equipment, and fine them tens of thousands of dollars.
There are two areas of concern regarding privacy and amateur radio, the FCC license database and encryption over radio.
Licensing with the FCC for amateur radio requires being entered in a publicly searchable database. This record will include your mailing address. And while your email address isn't shown on the website directly, it will appear in PDF forms that can be viewed by the public.
The privacy solution to this is to rent a Post Office box for the mailing address, and to create an email alias that forwards to your normal account.
Transmissions that are encrypted in whole or in part over amateur radio are illegal. All transmissions must be observable by third parties.
Licensing exams have generally been done in person, but some VECs are modernizing so that they can provide remote testing sessions. The ARRL website search is a bit more confusing to use, but is more comprehensive.
Locate a session using the sites below, and either check the VECs website or contact them to verify that the exam will take place, what the current fee is, and if you need to pre-register. They will ask which exam you will be taking during that exam session.