Any player you choose should have all the cabling and software you need to get the player hooked up to your PC.
The sites you go to for music will depend to some extent on which player you choose. Apple's iTunes offers files in AAC format, which players other than the iPod will not recognize (before someone who doesn't know tells you otherwise, the iPod does recognize and play mp3 files). Most of the other sites (Rhapsody, Wal Mart Music, Napster) offer files in a protected Windows Media format. The iPod will not play those files.
I buy most of my music from iTunes, but I do buy quite a bit from a site called emusic.com. They have files in mp3 format, but most of their stuff is from small labels and is definitely off the beaten path. They're great if you like less popular stuff like blues, jazz and classical. There's another mp3 site hosted in Russia that I've heard some good things about, but I haven't visited it so I just don't know. Someone who knows more than I will probably be happy to post the URL.
Generally, any of the sites where you pay for the music will be "safe", as in free of viruses and other malware. Since you're getting this for a teen (as in someone who doesn't have her own credit or debit card), you may want to look into sites where she can buy and redeem gift cards.
One thing about Napster - they have "unlimited downloads", but you don't actually own the music. You have to pay them a monthly fee (about $15 the last time I checked), or else your music will not play. I don't like that at all, but some people prefer it (I'm guessing that most of them are people who burn their music to CDs).
My suggestion - take your niece shopping with you when you're ready to buy the player, and take her someplace like Best Buy or Circuit City, which should have a good selection of various models. Let her try out all the players that fit your budget and pick the one she likes the best.