I think I'd like to become an astronomer. What degrees and training/experience do I need to become one? I've heard I should take science and mathematics, but I don't know which specific courses I should take and which degrees I would need to become an astronomer.
To get into scientific fields as a professional pretty much requires a doctorate these days. This can be in astronomy or your chosen specialty (see below), but many are also in physics; what the subject is called may depend on the institution you attend to get the Ph.D.
Realize that what you and the general public thinks of as "Astronomy" is likely not the same as what the professional scientific world calls "Astronomy". An "Astronomer" studies stars and groupings of stars. Someone who studies our Sun is usually called a "Solar Physicist" or something similar. Someone who studies other planets is a "Planetary Scientist"; this can be further subdivided into "Atmospheric Scientist" (which can be even further divided into "Atmospheric Chemist", "Atmospheric Dynamicist", etc.) and "Planetary Geologist", and so on. A scientist studying the origins of the universe is a "Cosmologist". If you have a particular area you are interested in, research this to find out what you need to learn to move into that field.
Regardless of the degree you pursue, you will need a lot of mathematics and science. During your undergraduate degree, I would recommend the following minimums: 3 semesters of Calculus, 1 semester of Ordinary Differential Equations, 1 semester of Linear Algebra (some schools combine ODE and LA), 3 semesters of Introductory Physics. I'd recommend a semester or two of introductory astronomy, although some schools don't consider that as absolutely necessary, surprisingly. Other useful undergraduate courses will depend on what you think you will want to specialize in, but would include more physics, more astronomy, chemistry, geology, meteorology, etc.
If you aren't yet in college, take all the mathematics and physical science courses you can in high school... unless you're interested in becoming a astrobiologist (someone who studies living organisms on other planets - a speculative field at this point, but it does exist), in which case you should get some biology courses too!
Learning computer programming would be good also, as these fields often require programming for data processing and simulations. Amongst general purpose languages, the Python programming language may be the most popular in scientific environments. Others used include IDL and MatLab. Linux and Mac computers are used more in the science world than in the broader populace.
As noted in Jeremy's answer and its comments, job prospects aren't great for astronomers and related fields. That said, I don't want to discourage you, just prepare you for reality.
Of course, as another answer noted, if you just want to observe the skies, no degree is required, just time and patience.
Well, if you want to become a professional astronomer, there is no obvious answer, since education systems can be quite different depending on your country. But astronomy & astrophysics is physics in the end, so a good training and strong knownledge in physics are definitely required (that involves math for sure). And in countries/universities where you can find an astronomy-oriented degree (at master level most of the time, before that it does not really make sense), you won't be surprised that most of the lectures are actually just physics or math.
Then, to get a PhD in the field is the way to go.
As for the positions, you can find some statistics in this paper released by ESO (the European Southern Observatory) for jobs in astronomy in the 2009/2010 hiring season. Bottom line is there were 65 opened positions worldwilde, that cannot be consider as "a lot".
There are degrees in astronomy but many astronomers have physics degrees. But there aren't a lot of positions going for professional astronomers (and those that are, you'd better be doing it for love, not money) so besides the degree, you're going to need to stand out from the crowd.
Or you could become what I prefer to call an 'independently funded astronomer' which doesn't require a degree at all, and you still get to do the stuff you love.