Looks like I can purchase a domain name for my website at Network Solutions for less than $13. I'm new to this so I have a couple questions if anyone has the answers...

If I purchase this domain name from them, can I use it anywhere? Like I have my website through Weebly.com right now.

Anyone know if they have extra fees or anything?

First the answer to your question is yes. You can purchase your domain name from one company and your hosting from another.
That said most hosting comanes also sell the domains. I use godaddy for both of mine because I can't beat the hosting price and stability. Chek and see if Weebly can also sell you the domain.

Network solutions tends to be the most expensive of all the domain and hosting services.

$13 is a little high, many real hosting companies can register your .com domain for $10 or less annually. One example is 1and1.com.

You can also buy your domain directly through weebly:
http://kb.weebly.com/buy-domain.html

Or set up a domain you buy somewhere else (although this is more complicated):
http://kb.weebly.com/domain-registrar.html

I use Weebly and although I bought my domain name from Go Daddy before I built the site, I know that yes, you can buy your domain from Weebly.

If you choose to buy your domain name somewhere else for whatever reason, there are no fees to use it for your Weebly site.... They have really helpful instructions on how to do it..
http://www.weebly.com/weebly/main.php

1and1.com is what my husband uses when he builds websites.

Thanks everyone! I decided to buy it from 1and1 instead. Now just trying to figure out how to connect it to my weebly website. I e-mailed 1and1 support asking them how and sent them the message that weebly says to send.