Thread type and brand for janome 350E

You aren't required to use one brand and/or kind. First thing you need to think about is Rayon vs. Polyester, I won't go into big explanations since you can find more info on the web, just use Google.

Next you need to choose a brand. There are tons of good brands and there are a lot of not so good brands. A while back I had the Bernina version of the machine you have (they were the same machine) and I had fine luck with a brand called Metro (metroemb.com) and the price was great. I now do embroidery as a business and have used several different brands to try to find what is best for us. Metro didn't run well at higher speeds on our six needle Brother machines. I will list some of the major players below that I know are considered the "main brands", I am sure I will forget some:

Madeira
Robison-Anton
Floriani
Fu-Fu's
Isacord
Marathon
Glide

I sold all my Metro thread with my machine otherwise I would have sent you a sample. I guess what I am saying is with my small home machine like yours I was able to get away with using cheap thread but now have to be a little more careful with my thread selection.

As far as where to look to buy thread, brothermall2.com has great pricing and low shipping. Almost everyone else I know of uses UPS and shipping is rather high for someone in the home market that may only buy a couple of things at a time. I think you can also buy Marathon thread at good rates without a business license. Not sure where you are but you can see products here marathonthread.com

Since you are asking, the next question might be what do I use now. Well right now I have a mix of Madeira, Robison-Anton and Marathon. Everyone will have their favorite and it may even change over time.

There are several guides out there that will tell you what kind of backing to use:
http://www.emblibrary.com/el/elprojects/holder.aspx?page=pr1153

On onesies we use cutaway and when finished we put "cloud cover stitch" (I think that is a brand name, we don't use any one particular brand though) over inside (backside against the skin) of the embroidery. This gives it a soft feel and is non-irritating.

Tees we also use cutaway.

Towels we use tearaway but we also use a water soluble topping on top of the towel. This keeps our top thread from sinking into towel and getting lost or having the towel all poke out the top of the embroidery. Water soluble is used on anything with a heavy nap that might cause your thread to disappear into what you are embroidering.

brothermall2.com has cheap shipping and has smaller portions of backing that may be more suited for the home market.

Absolutely, if you have any questions in particular just post them.