The AirPort Extreme will be fine for what you're doing. Typically you want your Server to run DNS for your services.
Setup your AirPort's DNS servers manually, point the first one at your Server's local IP address then the second at an outside DNS source such as Google, Open DNS, or your ISP's DNS servers. Then make sure your DNS service on your server is configured correctly for your Server's domain and has forwarding addresses setup.
This way, any of your devices on the local network will look at your Server for DNS first and any requests for domains outside of it's own will be passed on to the outside networks.
Once you're ready to have the services accessible from outside your network then look into a service like OpenDNS and possibly a service like DYN, to provide a dynamic DNS address if you don't have a static IP with your ISP.