Being a beginner at turning, I'd advise that you scrap the gouges until you gain some experience, especially with regard to turning bowls. It's so easy for the gouge to catch while turning bowls, and consequently damage the bowl or tools, or hurt yourself. Scrapers are the way to go until you find someone to show you the proper technique and to instruct you on which gouges to use on a bowl. You could try searching youtube for some videos on bowl turning and perhaps get the required knowledge there.
Take a look at
http://www.tinkerjohn.com , specifically his page on plans where you'll find a link to a youtube video on sharpening lathe tools using his homemade jig. You can get his plans for the jig for $5. He has a number of youtube videos on turning that should prove useful in gaining turning experience.
To begin turning, I'd try making some spindles. You can start with 1 1/2 x 1 1/2 x 8" blanks of clear wood ripped from 2x4s. Here is where you could use the gouges, as well as the scrapers, to gain some experience. But you should be aware that the edges of the various gouges (spindle, bowl, roughing) are all ground differently, and as the name implies, they perform different functions. Using the wrong gouge or using one improperly can lead to serious consequences as you found out. Catches on a spindle are much less severe than when turning a bowl.
Also be wary of the skew chisel. As a scraper, they're easy to use, but for shearing they too are tricky.
For a chuck, take a look at the Nova chucks (or similar style chucks from other makers like OneWay or Penn State Industries). The Nova requires an adapter that allows you to fit the chuck to a variety of lathes, including the shopsmith. If you change lathes, you simply buy another adapter. These chucks mount very close to the spindle nose, so you won't have that long extension you describe for your current chuck.
Don't forget to wear a face shield when turning.