Thanks for the advice, annieshane and Celeste! I have my friend's machine up and running now and it's great - I definitely owe them one for lending me it!
I'm not sure if it would manage quilting (it possibly could, as long as it goes very slow), but it's doing great for piecing. I think I want to try hand quilting to start with in any case.
I've tried to keep my hand piecing stitches at around 1/8" at the most in length, and have been taking a backstitch every inch or so to try and keep the seam strong. It sounds like it should be ok to wash though - I don't really want to wash quilts in the bath, if at all possible. I think I'll just need to try it in the machine on gentle and hope that everything goes well (and repair anything that doesn't!) :)
Thanks again for all the warm welcomes!