Well, the gist is - it depends on how picky you are about trails. It will almost always start to trail immediately, but it may not be noticeable until a certain point.

Additionally, forget normal exposure rules for astrophotography. It's generally about getting the most light in that you can. The odds of overexposure are pretty slim unless you're in a very light polluted area.

First off, your aperture will have pretty nothing to do with the amount of time before you get star trails. Its strictly a function of the rotation of the earth and the perceived trails is a function of your exposure time vs the focal length. The shorter the length, the longer you can track without noticeable trails.

• If you go ultrawide and don't really be too picky, folks have been known to go up to 90 seconds.
• At 50mm, you're really only talking about 18 seconds or so before they're noticeable (based on my own experience).
• At 300mm, you can get away with at most 5 seconds before trailing shows up.

These times will vary somewhat, as the apparent rotational movement of the sky varies between the poles and the equator.

If you want to track longer, you're going to need a tracking mount of some kind that follows the sidereal rotation. You can get a dedicated unit like an Astrotrac, you can piggy back on a telescope that already does tracking, or you can build a barndoor mount. See this question for more ideas there (stacking multiple images is covered there as well).

Other tips:

• You'll want the darkest sky you can imagine and make sure an point it away from the nearest city. That city 'glow' will show up in your pictures as light pollution.

• You'll want a large aperture - that way you can get more light in the short amount of time you have, but pick a sharp one. So if you've got a 50mm 1.8, you may want to stop down to 2.8 to increase the sharpness.

• Your next big challenge is focus. Its incredibly difficult to focus on distant stars. You have a few options, but your best choice is a Bahitov mask - mask in front of your lens that produces a particular diffraction pattern to help with focus. Check this question for more details.

• After than comes post processing. Thats a whole thing in itself. Generally, you'll be trying to remove light pollution and noise without removing stars, enhancing star / DSO (deep space objects - nebulas, clusters, etc) color, and push some faint stars / DSO into brighter existence.

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All this applies to falling stars as well, you just need to try to catch a star fall during one of your exposures.

There's also whole lots of general AP information in this question as well.