There are different rules for amateur rockets and space transportation.
Amateur rockets are unmanned rockets that have a combined impulse across all motors of at most 200,000 lb-sec and that cannot reach an altitude of over 150 km. These rockets are comparatively lightly regulated, but you need FAA approval to fly in controlled airspace (which includes all airspace above 18,000 feet) or near an airport. You cannot fly an amateur rocket on an orbital trajectory or into another country; for high-powered amateur rockets, you also can’t fly into clouds or areas with low visibility, have to stay far away from any person or property not associated with the launch, and have to fly during the day unless you get FAA approval.
If you want a higher-powered rocket or want to make an orbital flight, you instead have to meet the requirements for space travel. Space launch licenses are much more in-depth than FAA approvals for amateur rocket operations. If you’re launching a reusable suborbital rocket, you might be able to get an experimental permit. If not, you’re held to the same license requirements as a company like SpaceX. These requirements are substantial. Not only does the launch need to be licensed, but so does any permanent launch site. The launch needs to be reviewed for policy (does it affect US national security, foreign policy, or compliance with international agreements); safety; payload acceptability (both for policy and safety); financial responsibility; and environmental impact. Experimental rockets have streamlined procedures (they rely on the rocket staying within the permitted operating area to ensure safety), but they do still have to meet policy, safety, financial responsibility, and environmental requirements.