I highly recommend getting a used machine from the late 70s or early 80s, a Singer or White brand if you can find one. Newer machines have more plastic parts, which can break if your thread tension is off. You can avoid that by buying a used older model. Search your local craigslist.org or freecycle.org for deals!
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I learned on an old Singer. Reason is, the folks who made them weren't sophisticated, either. In less-than-modern times, there were fewer gizmos to diddle on the sewing machines. So my advice would be, check out a goodwill store, or maybe even an antique store, and find either an old electric or maybe (if YOU can figure out how to work it!) a foot-operated sewing machine. Just thread the needle from the spool on top and go after it.
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The Janome Hello Kitty is a great one! Simple to use, cute looking, and definitely a conversation piece, yet more so it's simple for beginners.

Check out all the great reviews here:

http://wize.com/sewing-machines/p209561-janome-hello-kitty-11706#t=587

"Light weight and portable. Very distinguished green that nobody else will have. (It could of been pink!) Excellent for the beginner or someone who just wants A sewing machine. Thread tension can be a pain at times."

"Instructions to wind the bobbin and thread the machine are given in the illustrated manual. It's so lightweight and portable. Perfect for teaching children or beginners to sew. I got this for my daughters ages 6 and 10 and they love it!…"
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No 1 Brother LS-2125 Mechanical
Source(s):
http://wize.com/sewing-machines/t587-beginners

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I'd say a Brother. You should be able to find one on sale for around $99.
Source(s):
I have been sewing for nearly 40 yrs. (alterations/drapery/decor) I've owned many machines. Singer technology was sold and resold several times over the past 18 yrs or so...(furnature devision let go entirely) the sewing machine end of Singer sort of got lost. (that's why so many Singer stores disappeared) Nearly all less expensive machines are made as "throw away" when they go bad..BUT I owned a little Brother machine for about 10 yrs. I used and used it...In fact I used it for about 18 months when I because a sub-contractor doing alterations for a shop in my town. I was able to hem jeans WITH that heavy thread on that Brother. ( i can give you a tip on how to do that) ;o) I would say... I literally "dogged" that machine...using it for business and did aprox $30,000 worth of alterations on that thing... ;o) Some kid came in the shop..and pullled hard on that tension wire...nearly pulling it out... Being a sealed unit.. it could not be replaced...BUT THE MACHINE KEPT SEWING ANYWAY!!! AMAZING...

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