The first American moon launches were in the ranger program. Ranger 3 and 5 both missed the moon in their 1962 attempts. Where are they now?
To add some numbers to Ranger 3 and 5 orbits, from RAE Table of Earth Satellites (DOC) for artificial objects launched in 1962 (Source: SATLIST, Ruimtevaart Filatelie Club Nederland), last revised in 2001:
Atlas 121 D - Agena B 6003 primary payload (Cospar ID: 1962 01 A, NORAD No. 00221), a hexagonal cylinder 2.51 m long, 1.52 m in diameter and 390 kg in weight (wet mass), passed 36,800 km ahead of the Moon on Jan. 28, 1962 23:45 UTC and is now in a 0.985 x 1.164 AU heliocentric orbit with orbital eccentricity of 0.083 and 0.40° inclination to the ecliptic. Its orbital period around the Sun is roughly 406 days.
Ranger 3 Agena B stage (Cospar ID: 1962 01 B, NORAD No. 00222), a 6.7 m long and 1.52 m diameter cylinder with dry mass of roughly 770 kg is in unknown heliocentric orbit.
Atlas 215 D - Agena B 6005 primary payload (Cospar ID 1962 55 A, NORAD No. 00439), a hexagonal cylinder 2.51 m long, 1.52 m in diameter and 342 kg in weight (wet mass), passed 735 km behind the Moon on Oct. 21, 1962 15:50 UTC and is now in a 0.951 x 1.064 AU heliocentric orbit with orbital eccentricity of 0.056 and 0.41° inclination to the ecliptic. Its orbital period around the Sun is roughly 369 days.
Ranger 5 Agena B stage (Cospar ID 1962 55 B, NORAD No. 00440) is in unknown heliocentric orbit.
It depends on how fast you go.
When you exceed Earth escape velocity, you will leave Earth orbit and end up in an orbit around the Sun, similar to the orbit of Earth. That's what happened to Ranger 3 and 5.
When you do not exceed Earth escape velocity, you will end up on a highly eccentric Earth orbit (very high apogee, very low perigee). When the craft does a direct-ascent (in contrast to entering a parking orbit around Earth first) the perigee will likely be below the atmosphere or even below the Earth's surface - in other words, the craft falls back to Earth and crashes (if it survives atmospheric reentry).
But keep in mind that the Moon's gravity acts on the craft during a fly-by. Depending on how closely the craft passes the Moon and on what side, the Moon's gravity will alter its trajectory. This can give the craft an additional gravity slingshot sending it onto an Earth escape trajectory, change a crashing orbit into a surviving Earth orbit or change a surviving orbit into a crashing orbit. But a surviving orbit will not be stable. In any case, the orbit will cross the Moon's orbit again, so at some points in the future it will make additional Moon fly-bys which will again alter its orbit. So at some point in the future the craft will crash either into the Moon or into the Earth or get sent onto an escape trajectory.