how do space ships come to rest when they get into space?
Rest with respect to what?
Are you thinking of geo-stationary orbits ? Or 'entering' the L4/L5 halo positions ? Taking up Mercury or Mars orbit ??
To 'get there', you need relative velocity, then you must shed that excess velocity with Retro-rockets or whatever you want to call the payload's final propulsion system. To minimise fuel use, spacecraft may use gravity assists, a diminishing succession of elliptical orbits with sparing thrust at critical time, ion thruster or aero-braking...
A really careful trajectory that uses planetary gravity to sling-shot probes can easily double or treble the available payload...
The short answer is that they don't really come to rest when in space. Like roller coasters without brakes they are (by intention of the designers) always moving (actually, free falling) with fairly great speed "around" the body they orbit so that they won't "fall into" the body.
In case you are thinking of geo-stationary satellites, as Nik mentions, you should know that such satellites are in fact orbiting the Earth, they just do so at exactly the same rate as the earth rotates, so when you view one from the surface of Earth the satellite appears to just hang there (see ). Its like a child sitting on a merry-go-round watching her (fairly athletic) dad running along on the ground just outside with his video camera keeping up with the rotation of the merry-go-round; the man is not moving relative to the view of his child even though he is running like mad over the ground.
with respect to the moon.
welcome to pf!
hi emma! welcome to pf!
the usually fire their rockets "backwards" on the way to the Moon, to reduce their speed relative to the Moon, and then again on the other side of the Moon, to make the orbit circular
see eg http://en.wikipedia.org/wiki/Apollo_8#Lunar_sphere_of_influence"
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