Congratulations on such a wonderful surprise! I embroider on my Bernina 200E upgraded to the 730E and it took me quite a while to learn all the ins and outs of good embroidery and I am still learning. I use mine for many things beyond embroidery motifs, such as quilting and even even using heavier cords with that special cording foot Bernina makes.
As for stabilizers, I have given up on tear aways except in rare cases when I use them in bags or where I am not going to wash the item I am embroidering. I find it often pulls the stitches out of whack even when you are careful. Instead I use wash aways. I particularly like the kind that has some substance to it rather than the clear plastic type, though that's useful for toppers when you need to sandwich your embroidery. The clear stuff gets stiff when left out, so if you do use that be sure to keep it well sealed in a plastic bag or container. For most applications, I use Stabilize It, which comes in a variety of weights. I buy it online at Nancy's Notions. Here's a link:http://www.nancysnotions.com/product/wa ... +sheets.do
I have found it the most economical and coincidentally the best stabilizer for most things--a nice combination. There are situations where you might want to use the sticky washaways. This would be when you don't want to or can't quite get your item in the hoop, so you hoop the sticky stuff, score and remove the top of the cover sheet and stick your item directly onto the hoop in exactly the right place. This is a little pricey so I only use it rarely, but it really is great for using your embroidery system for quilting.
I rarely embroider directly to my item, but I have done with great success. I usually use either bridal veilling or organza and stabilizer, embroider on that and then applique it to the item. (I'm using "item" because I embroider a lot of things...quilts, bags, ready-made clothes, etc.)
If you need to embroider directly onto your item because the design isn't independent, then be sure to stabilize it well and do a practice run.
One of the things that makes the most difference is ensuring you are using the right threads and needles. I use mostly either Superior threads (see link on this site) or Isacord threads from this site: http://www.embroideryonline.com/index.aspx
I find 40 wt makes a good size for most quilting motifs. If you are using quilting patterns, then you would want to use whatever you use for any quilting.
Also be careful in your pressing following the embroidery. Be sure to use a pressing cloth or press upside down onto a padded surface.
Like I said, there's a lot to learn. Hope that helps in answering a few of your questions. Did your machine come with lessons? If it did, I encourage you to take advantage of them. They help a lot.