One of the great things about astronomy is that it's really one of the few fields of science where amateurs can still have a large impact. Most nearby supernovae are actually discovered by amateurs, not professionals (including one just a few days ago [1]). In fact, the Japanese amateurs are famous for discovering supernovae because they get to catch all the ones that go off when it's night across the Pacific Ocean. But they also have collected a lot of very good data on variable stars, as well. [2]
Part of the reason amateurs have been able to play such a big role is that traditionally astronomers have used their resources to harness very large telescopes to study small patches of the sky very deeply. Study of bright objects across large regions of the sky only requires small telescopes, so it wasn't really worth professional astronomers' time to conduct those observations. This has been changing lately with the rise of robotic telescopes, however. Now that professional astronomers don't have to operate the telescopes themselves, they can collect bright transient data across wide swathes of the sky. [3]
To aspiring amateur astronomers: you don't even need an expensive telescope or clear skies to explore the night. iTelescope [1] lets you rent time on telescopes around the world and is surprisingly easy to use after learning the interface. I've used it to take pictures of galaxies and other objects of interest. You can even write scripts that point the telescope and have images delivered to your email.
In fact, asteroid discovery is well within the purview of amateur astronomers [2]. Lots of interesting astronomical phenomena are observable by amateurs these days.
(I'm not affiliated with iTelescope, I just like it)
Yeah but kind of expensive... I believe the best combination would be a small telescope to learn the basics of the sky and then if you want to go deep, rent the iTelescope service.
A small telescope is actually not the first choice for amateur astronomy. Pick up a pair of relatively cheap 7x50 or 10x50 binoculars (sizable but not enormous), and a good blanket. Do not try to hold the binoculars in your hand - lie on your back with your head firmly against the ground and rest them on your eye sockets (binocular hand-shake is the main reason people dislike them).
These are more portable, more user-friendly, more comfortable than any telescope. They're much better put together than the sparse selection of cheap telescopes at this price level. $50-$100
After this, I would consider an 8" dobsonian manual telescope, or a motorized alt-az goto scope with mount in the 4.5-6" range. $500-1000
A serious (but still amateur-level) 8" astrophotography setup (mount, sensor, telescope, filters) that will take the kind of nebula pictures people see coming from the HST, iTelescope, or Slooh, is going to run $5000-$10000.
I had certainly not heard of Isaac Roberts before today. This was interesting. Years ago I began experimenting with astrophotography using my 8" Meade Schmidt–Cassegrain telescope. I actually used a cheap webcam with the lens removed. I got the idea from http://www.cloudynights.com/item.php?item_id=141.
Part of the reason amateurs have been able to play such a big role is that traditionally astronomers have used their resources to harness very large telescopes to study small patches of the sky very deeply. Study of bright objects across large regions of the sky only requires small telescopes, so it wasn't really worth professional astronomers' time to conduct those observations. This has been changing lately with the rise of robotic telescopes, however. Now that professional astronomers don't have to operate the telescopes themselves, they can collect bright transient data across wide swathes of the sky. [3]
[1] http://www.bbc.com/news/world-europe-27185965
[2] http://www.aavso.org/
[3] e.g., http://www.astronomy.ohio-state.edu/~assassin/index.shtml