When I started, my first idea was to practice on the cheapest material I could find, so that if I messed up, it wouldn't hit my pocket too hard. That's OK in theory, but the cheapest stock also produces the poorest results - which can be really frustrating when you're trying to get a good finish with the tools.
If you're experimenting to see what the tools can do, cheap is still OK, as long as you don't expect a good finish. Turning green timber is often recommended too, as it's kinder to both the tools and the turner, although I prefer seasoned stock.
For spindles, try some sapele - it's easy to get, relatively inexpensive, but it is fairly open-grained, so requires a bit of care and good sharp edges on the tools. If you get a dig-in, it's not quite such a shock as with, for example, oak.
For bowls, sycamore is hard to beat - comes off in nice long ribbons with a good edge, easy to sand and finish, readily available and not too expensive. If you can find some Cedar of Lebanon, then you not only get a great turning material, but a lovely aroma whilst you're working - it's my absolute favourite.
We all have our favourites and preferences and turning seems to bring this out quite readily. I've had good results with utile and zebrano - purpleheart too, whereas other turners I know detest them all.
If you wanted to try something unusual, you could get a banksia nut - won't break the bank, but it's weird stuff to mount. HTH
Ray.