In any woodturning project, you will start out with a "blank". Those can often be quite small and reclaimed from (free) scrap wood! Some even literally grow on trees, but often you'll need to make some preparations before actually starting to turn.
In general, you should look for an even grain structure and avoid knots for easier turning. Single flawless pieces are optimal, but with some experience you can also try to fill cracks with a suitable resin or turn carefully arranged & glued "segmented" blanks. Whatever you end up doing, make sure to estimate the center of the blank as closely as you can to reduce unwanted forces and unneccessary waste of material!
The more unneeded wood you can remove before turning, the better! Octagonal blanks are relatively easy to make in most cases, and the extra sawing steps are quickly worth it at larger diameters.
Contrary to many other woodworking techniques, a bit how much? of moisture (or even "green" wood) can be helpful when turning. Very dry, hard woods (e.g. beech table leg used for some Tigerente wheels) can be hard to cut and produces small chips rather than nice, smooth shavings.
Very soft wood, like fir/spruce, is actually NOT easy to turn, you'll often get tear-out instead of nicely shaven surfaces (similar to planing these woods). So while it's good to start "soft" to understand catches & tool angles, prepare yourself to acquire some higher-quality wood for your take-home projects.
Micha has tried plane (Platane) and hated it, but is often seen in the Space happily turning maple.
Spindle blanks (Kanteln), i.e. those you would turn lengthwise (and often between centers), ideally have a straight grain along the spindle.
The diameter of these is often quite small and you'll get away with a square blank.
Tree branches and small logs also make great spindle blanks if they're not too curvy!
Cutting Square Blanks
step 1: use the narrower width of your stock to set the table saw's parallel fence
Glued Spindle Blanks
When your stock is too thin for the work you have planned, you can glue two pieces together - just make sure to have them in the right orientation so the bond won't crack due to the wood "working" later on!
Very schematic grain constellations that should work: 
- () (might open in the center, but not visibly)
- NOT )(
- NOT =/ (or any other mix of different cuts)
Have a look at the illustrations on HolzWerken.net if this doesn't make sense!
We do not have a wood lathe capable of serious bowl turning yet! But perhaps we can already collect some information anyway...
If you do not want to make your own blanks or, perhaps more likely, don't have the proper kind of wood at hand, you can buy blanks at numerous stores. Just do a quick online search for "Drechselholz", "Drechsel Kantel" or "Drechsel Rohling" and you should find what you need! As usual, please share your experience if you try out a shop :-)