In the beginning it is usually beneficial to use something like dyndns and point a dns name at your home router and use port forwarding to direct game traffic to a server in your home. Why pay someone for something you already pay for? Many home wireless routers even have ACL (access control lists) functionality now so you can allow only certain networks or individual IPs to connect to your game instead of opening it up to the internet. If your existing router setup doesn't support ACLs, routers that do aren't very expensive (sub $100 US for lower end models).

As things get moving you may want to look at virtual private server hosting (google VPS hosting) which is a cost effective way to use someone else's network and resources. The added benefit here is that they manage the power and internet connections so even if your home loses power or internet then your game server is still up.

The VPS environment will be good for quite a while (depending on the provider) as they usually have the ability to scale available hardware, number of nodes, bandwidth, etc. Bandwidth may become expensive if you don't keep an eye on it but hopefully at some point you can start to recoup those costs through subscriptions to your game or some other means - donations, advertising, selling out to a larger game company...

You don't need any programs for hosting, you just need to know how to configure your firewalls (router firewall and PC firewall). Sometimes its called virtual server on the router.

On my router I just add a virtual server. Set options like the local server's IP address and any ports which are going to be used. I'm not that knowledgeable in networking but after that i'd assume the router would start routing packets of the port to that IP address specified, how else would it work?