BANG!
Plain jane model 870 Remington. Reliable and rugged and won't break the bankbook.
You can get a great used Remington 870 for under $250. (I have no clue what they cost new) In a short barrel, there is no better home defense weapon.
If you are quick and very good, you could shoot clay pigeons with it.
I know I could hit clay all day with mine, but it would scare the heck out of anyone standing close by.
You got to hit them almost as soon as they come out!!
But seriously, ther are several very good shotgun with multible barrels out there that would work well.
For home defense, the 870 is king.
My first shotgun was from Sears Roebuck! If I remember correctly it was @$100.00. I still have it, looks great and shoots great. 12 guage pump, shoots up to a 3in Magnum. Works great for me. Keep it clean and well lubricated.
Use it for small game, rabbit. pheasant, etc. Have also used it for White Tail Dear, using slugs. This was in Virginia and Rifles were not allowed in that area. Of course it's very easy to get an Irish setter confused with Deer in Virginia! Some of those great deer hunters we all know hunt that breed of deer with rope burns around their neck!!
A Remington 1100 would be easier for your daughter to handle, less recoil.
Shotgun ping
I'm also in the 12 gauge 870 crowd. I found mine for $100 but the barrel turned out to be bent, right at the 18" spot, so it's a riot gun now. Extended mag, etc. The only way it can be called a game gun is if terrorists are a protected species.
Get your daughter a 20 gauge, if she's big enough, .410 if not.
And an automatic like the 1100 would be more suitable for trap or skeet. 12 gauge, 28 inch barrel, screw in chokes.
For sure, get a 12 guage. You can buy "low recoil" loads if recoil is a problem for your daughter, and you can use full power loads. I'd recommend a pump action from one of the major brands.
450 Bucks or less if ya have a FFL Gunslinger Friend. It's all ya need for multipurpose self defense as is yet I added ghost ring sights and a reciever rail for my old PVS 4 night vision scope. Good night fighter tool when accessorized with 3 large Rottweillers....
There are two big schools of thought, but I will say this.
GO with a 12 gauge. You can buy or reload shells that will not pack enough kick to hurt your 12 yr old daughter, and besides, if you are going to the range, you don't want to slam yourself with it.
Secondly, you have two options: pump or auto. The pump is cheaper, easier to clean, and simpler to operate and repair jams. The drawback: harsher recoil.
The auto is more expensive, a bit more challenging to clean (more moving parts) and jams can be a bit more difficult to work around. On the other hand, the recoil is reduced even with less powder shells but you take the chance of loosing the energy to eject spent hulls.
Regardless, for a 500 dollar price tag, a Remington 870 or 1100 is the way to go. You can find them at WalMart, pawn shop or gun store. Be wary of used guns unless you have a gunsmith to take it to and have it thoroughly checked over or if you know what you are doing when you select one.
If you are going to use this rifle for home defense, you will want two things: 1) tritium sights (the sights glow in the dark due to the tritium isotope in them-- you don't have to "charge them up" with light), and 2) a light for target ID.
Without those two items, your shotgun is not going to be very useful at night.
You'll want something that is easily accessorized and has plentiful spare parts. IMHO, a Remington shotgun is #1 in both categories.
You probably want a shortned stock (sometimes called a youth stock) because the standard stocks are quite long.
For a light, go with a Surefire (they are the only manufacturer I would trust to make a light for a shotgun) that has a disable switch so that you can avoid accidental light discharges.
In closing, I'd recommend you consider the following (I don't have this one, but will probably purchase one next year):
Wilson Combat Standard Model #SSTD-AS
If you scroll down on that website, it's the second one from the top. It has a light, a tritium front sight, and shortened stock. But it's pricey... about $1165.
What do you plan to do with it?
I went out for early dove season this morning...took the limit. :)
http://www.spartangunworks.com/spr310_field.htm
I'm shooting one of these now. It's under $500, comes with 4 different choke tubes and you can buy more, shoots nice, isn't too heavy, absorbs recoil well, and it's built like a tank. The wood is nice, and the wood/metal fit is good, too. It's made by Baikal in Russia and is imported by EAA. I like it a lot, so far.
"Big 5" sporting goods carries it, they have stores in a lot of places. Don't get the Turk Khan which is a little cheaper, and real nose-heavy.
I'm a righty but I'm left eye dominant, so I had to get the stock bent to straight by a gunsmith since I shoot shotgun lefty. I'm not sure if they are all cast to the right, or just mine.
For skeet, buy 1 oz loads. 1 1/4 oz of lead kicks a little more, and isn't more likely to break clays.
Also don't shy away from lessons from a NSCA certified instructor, for your daughter either. I shot OK before, but after a few lessons from a good instructor, my shooting improved a huge amount.