@Syamlal: BMXer? Awesome! So you no doubt have some skills :)
Few things to clarify here
Firstly on the term - DH bike. A bike that is ridden downhill is no more a Downhill bike than a bike that is ridden on the road is a 'roadbike'. You can ride a full-sus XC bike downhill but that doesn't make it a downhill bike.
Not all full-suspension bikes are downhill bikes. There are many kinds of full-sus bikes which are categorised in ascending order mostly based on the amount of travel the suspension affords - XC, Trail, All-Mountain, Enduro, Freeride and Downhill. The downhill bikes are the Big Poppa Pumps of the full-sus world. Unfortunately, there really is no such thing as a 'beginner downhill bike' in the sense of cost. The entry-level DH bike is really not much cheaper than a 'pro' bike because its similar design-wise but just has slightly cheaper components on it.
Its also because nobody really learns on a DH bike. People learn with shorter travel bikes and then graduate to full-on DH bikes. So what you probably need is an All-Mountain bike. Even a full-sus XC bike will simply not do. Why? Because the geometry of a bike designed to descend is very different from something designed to ride cross-country.
Now, the unfortunate part is that a true All-Mountain bike is also pretty expensive because it has to be a jack-of-all-trades i.e. it has to allow you to pedal it uphill and then allow you to take on a downhill track.
This requires quality engineering and componentry and won't be cheap because of that old biking rule - light, cheap, strong - pick two of the three. If a bike is cheap and strong, it won't be light. If it is strong and light, it won't be cheap. And if it is cheap and light, it won't be strong.
Good luck :)