I'm a born-again sewer too. I've only ever followed one pattern and that was for a high school sewing class. I've never had a machine but I just bought one for myself the other day. The simple, no pattern projects I'm starting with are:
-catnip filled square sacks/ pillows (the cats LOVE these, I swear my cats are catnip junkies)
Then I'm going to do a bigger, major project like a cat bed.
-sissors (I read you can sharpen them by cutting up a fine grain piece of sand paper)
-seam ripper (yeah, I used it a couple times on my apron when I accidentally sewed over 3 pieces of fabric instead of 2, heh. These come in REAL handy)
-multiple sized needles (I bought this cool multi-colored thread and realized when I got home that it required a heavy needle, luckily my sewing machine came with 2 of each size needle!)
-pins and pin cushin (I'm currently using a little stuffed animal but I plan to make one of those wrist cushins)
Since you're a new sewer, I'll point out that you need to buy your needles and bobbins for your particular machine. They have numbers that correlate. (This is one of those things sales people forget to tell you and then you waste a bunch of money on the wrong type needles).
From my own search on these forums about newbie sewing I found that there are lots of easy patterns and they often go on sale for a dollar or two. Then I went to the store and sure enough, lots of easy patterns, but they weren't on sale that day The hard ones were on sale. The easy patterns are marked "easy" or "sew easy" or "basic" or something like that.