NetworkSolutions.com is the government contractor that maintains who owns what domain name.
So the best thing to do is register your domain with them. If you do the thing where someone else registers it for you, then they MIGHT be registering it in their name, so if you later decide to leave their company, you can't take your .com name because they actually own it.
Personally, I like and use iPowerWeb.com. They cost about $100 for a year, but give you lots of space and good service. Many of the cheaper web service providers are pretty shady and not very good.
obviously you should read the terms of service of any domain name provider and which webhosting plan works best for you depends on what you need it for. doteasy maintains that YOU are the sole owner of any domain name registered through them. personally i think network solutions and ipowerweb are overpriced, but then budget is a big concern for me, and less of a concern for others. i also like that i can get my domain and hosting at the same place and have all these easy to install plugins just waiting to be copied and pasted.