Richy;
First question I will ask is this: Is your daughter really serious and committed to singing, do you see her sticking at it long-term? Two or more years?
If you can answer yes to that, then I would strongly suggest you look at purchasing something like the Focusrite Scarlett USB Audio interface 'Studio Pack'. This will give you the audio interface, a pair of headphones (which sound nice!), a CM-25 condenser microphone and all the cables you need to get going. The mic is not as durable as the likes of the Shure SM-58, but as you will be placing it on a mic stand this should not be a problem.
Not sure what country you are in, so I will use Australian dollars as a point of reference. The Focusrite Scarlett Solo "Studio pack" is around $320-350 AUD as a guide.
There is only one caveat with the solo, it has unbalanced (RCA) outputs, so if in the future you decide to upgrade to some monitor, like what I have, the Alesis Elevate 6's which are balanced input, you could have problems with earth-loops. (Before anyone says, yes I know the Elevates have unbalanced RCA in's, but this doesn't stop earth-loops). If you are monitoring via a domestic amp/speakers then this is not going to be an issue for you.
Think of this. Buy the SM-58, the interface, leads, boom stand - this could be in the region of $300 - so for an extra few dollars (with the Scarlett) you will have something that she can use and work with for many years and grow with. For recording software (if you are going computer based recording), then Audacity will fill the bill - it is free! It is also easy to use for recording single tracks at a time. It will also do multi-track for when she gets to that point.
By the way, the SM-58 is a balanced output, so you can not just 'plug it in' to the home stereo - you will need a pre-amp, that's where the audio interface box will come in.
In my mind, I would start with the Focusrite Scarlett Solo studio pack, it will give you all you need, team it with Audacity and you have a pretty neat system. Sadly, I gave my Solo to a mate in the US as I have upgraded to a 2i2, I would have been more than happy to send it to her at no cost.