I use half-hard wire in sterling silver or gold-filled. Dead soft is trickier to work with in that the shape deforms more easily. I would say, get comfortable with using half hard first.
The gauge depends on what you are doing with it. The lower the gauge number, the thicker the wire-- size 18 or 20 gauge can be used to make clasps, size 24 gauge (the thickness of most head pins) is good for making dangle earrings in fully-drilled pearls, size 26 gauge is finer and can fit in tiny drill holes in small gemstone beads, etc.
Don't start off using the more expensive wires...you will waste money. Buy inexpensive brass/copper wire and practice all your designs with that. You won't be afraid to make mistakes since the wire is cheap, and will learn faster. I still do this when trying to do something new. However, you will find that the brass/copper wire is softer than gold-filled and sterling while base metal head pins are significantly harder to work with than the same gauge gold-filled or silver head pins.
You can work-harden (and straighten) your wire by running it through nylon-head pliers, or by hammering it with a nylon-head mallet on an anvil. This will make it denser, stronger and more springy (whereas hammering it with a chasing hammer will make it flatter, more rigid and more brittle.) Never hammer over wrapped or crossed wires as they will break.
You will need the right tools-- definitely get round-nose pliers, some chain-nose pliers and I also like a pair of bent-nose pliers for getting into tight spaces. You will also need wire cutters-- I use a side-cutter (there are also end-cutters.) If you are making several loops and want all of them to be the same size, make a mark with a permanent marker on the round-nose pliers so you know exactly where to place the wire along the taper. Otherwise you may end up with one loop larger than the other!
I agree with GemGeek about looking online for videos-- there are excellent ones on YouTube.