I would spend at the very least $50.
Good brands are Wilson, Head, Prince, and Babolat.
Look for racquets where the frame is one solid piece. Cheaper racquets often have a two piece frame. You can see the separation of the two frame pieces at the lower part of the head, where the strings attach at the bottom of the head.
Grip size is extremely important. If you have a small hand you will likely want a grip around 4 1/4" - 4 3/8". Larger hands will use 4 1/2 - 4 5/8".
Don't get the wrong size grip, or it will make the racquet difficult to hold.
Also, if you are very small in height, they make shorter length racquets for little people. They are 26.5" long, and you can find good quality models
in this shorter length.
If you are thinking about spending a little more for a better racquet, ask the store if they have a demo program. Some stores have demos they can lend you to try them out before you buy them.
I also recommend finding a practice tennis wall in your neighborhood that you can spend time working on your strokes. Some public tennis courts have practice walls. Also, schoolyards sometimes have big walls.
If you decide to try to find a decent used one on Craigslist, you might get a very good deal. Check the frame of a used racquet carefully to make sure it has no cracks.
Many times when racquets become discontinued, they drop the prices down significantly in stores like Big 5. You can sometimes get a racquet that previously sold for $200 for around $60 - $80.