Lots of good advice already - pulling in a couple different directions, so why not add mine? ;-)
Two daughters, now 14 & 11. I've been 'into' cycling since I was very small (early '70's) - got my first 'cool' bike in '80 (Huffy Mono-Shock - you don't even know heavy, or the fun of having the skin ripped off your thigh, until you bottom out one of these) my first road bike in '82 (Huffy Santa Fe, 24". Won it in a read athon). Discovered Mountain biking in college in '87 - maxed out 3 cards to buy a Raleigh Chill. bio-pace baby.
So I dig bikes. Garage is full of them - Raleigh, Trek, Diamond Back, Specialized, Kawasaki.... and I'm not afraid to spend money on them.
So we went this way:
1st bike: Barbie bike from Target for the first one, Little Mermaid bike for the second. Same bike, different stickers. I wish I'd known about the striders at the time, would have definitely gone that way. But - I took a golf lesson once (didn't really stick) but one thing the pro told me did stick in my mind. When you are learning, your equipment should call you from the garage. Get whatever clubs you think are cool - if you get good, get good clubs.
Same thing with the bikes - both girls thought their first bikes were AWESOME and wanted to ride them all the time. Cheap buy in cost - a couple hours (no lie, hours) making the brakes actually function on each - and they rode those bikes until the training wheels fell off.
Second round was craigslist - garage sales all the way. For one of them I was in an 'unemployed' time period when her birthday rolled around, so I hit a bunch of garage sales and was actually given 2 20" 'mountain bikes' for free - neither worked. Another couple hours of part swapping - and daughter #1 couldn't have been happier. She cleaned that thing as much as she rode it, she loved it so much.
2nd daughter got a $20 garage sale one when she moved up in size. And so it went. Until they hit the 26" wheel size.
One got a new Raleigh (bought on closeout from REI) the other inherited my wife's Trek 4900. Both bikes are super light, well built and will last the rest of their lives, if they want them that long. The teenager has already moved on to a 3 speed cruiser - much to my sorrow.
So that's how we did it - cheap bikes that they liked when they started, to get them hooked. Cheaper bikes when they grew, didn't really care if they liked them or not, as they were going through the sizes - and then reasonably inexpensive 'real' bikes when they hit grownup size.
Worked for us - YMMV.