You can get great beginner machines for well under your price limit.
Singer and Brother are the leading brands for this market. They offer lightweight, portable machines and most of their models receive excellent reviews, even from new sewers.
I'd say think about how far you plan on going with sewing. Also consider how much freedom for experimentation you'd like to have. By this I mean, do you think you'll try out many different types of fabrics and create many types of clothes for your daughter?
If this is the case, I think a Brother CS6000i would be good for you. It has lots of different types of stitches and it's computerized, meaning that selecting stitch types is as simple as pressing a button. With more stitches, you'll be able to experiment with more types of clothing. This machine will cost around $150.
If you really just want a bare bones machine, there's the Brother LS2125I with only 10 basic stitches. Like the cs6000i, it's also lightweight and made out of plastic. You can get a new one for around $75.
Singers are for people who want a "heavy duty" machine. These machines are made out of metal (thus they weigh a little bit more) and have high speed motors. They have plastic machines too, but they don't seem to be rated as well as the Brother machines.
I bought my mother a cs6000i as a birthday present, she's been wanting one for a while, and so far she loves it. That model has over 500+ reviews on Amazon, most of them very good, so I thought it would be a good choice. So far, it is.
Hope this helps.