Heck, we catch something going thousands of miles an hour every time we go to the ISS which is travelling at 17,200mph. Been doing that sort of thing since Gemini VI and VII rendezvoused in 1965. It's usually a case of matching orbit, then increasing orbital velocity whilst maintaining a stable orbit (which involves thrusting at your apoapsis and periapsis) until you creep up on the object, then decrease your orbital speed until you match velocities, then thrust at the object and increase and decrease your speed to meet it.
It's a bloody hard thing to work out when you first time you try it but when you get your head around it, it's not too bad.
The kicker is having everything up there in order to resupply it all, once we get a cheap and reusable way to orbit then it will get a lot easier.
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