Congratulations! You will have your hands full with babies, so I hope you can sneak in some time to play with your new "baby" embroidery machine.
You CAN use any thread,it makes nice candlewicking results, but you will be much happier with the results of regular embroidery designs if you use embroidery thread. It comes in rayon or polyester, has a nice sheen, and is thinner than sewing or serger thread. if you want a dull finish, you can use serger thread. You will need bobbin thread (black and white) which is even thinner. I use pre-wound bobbins but you can buy a spool and wind your own. If you do fsl-free standing lace (items are made on a wash away stabilizer and only the thread is left, lots of fun projects) then you would use a matching bobbin of the embroidery thread color on top.
Stabilizer does just that, stabilizes your project. You almost can't over stabilize, but you can under stabilize. If you do not use it, or enough of it, you will get puckers and other issues. Tear away works for lots of projects, cut away is better for t-shirts and knits, as it needs to stay in there when you are finished to "stabilize" the knit. Looks wonky when it's washed without proper backing. I would certainly have some thin, wash away "topper" (looks like saran wrap) for topping towels and t-shirts (keeps the stitches from sinking down and getting lost in the nap), some "vilene" cloth-looking wash away for fsl. There are heat away and sticky-backed, and lots of others than you can add at a later date.
Have plenty of needles available, you will change them more often than when sewing. Needles can be the one thing that is making the machine act funny, and when you put in a new one, all is well.
There are methods of embroidery where you don't actually hoop your item (like a small onsey) and you stick it down to stick stabilizer, or spray with a temporary adhesive (make sure it is for sewing, gums up the needle if it isn't) or pin it down to your stabilizer, so lots of tricks to learn.
I personally would not purchase my supplies at a craft store. Jo-anns, Hancocks or your local machine dealer is a better bet, best prices are on-line. So very many, can't begin to list suppliers.
This is not an inexpensive hobby, and it is addictive! One spool of blue isn't enough, you will want 16 shades of blue, and green and yellow, lol. Hope this helps.