So I presume then that you will use the router with a modem to provide a connection to the internet and then all computers connect via the router to the internet rather than there being a 'host' computer that shares it's internet access.
If this is the case then all computers will have simultaneous internet access, although I'm not sure how much of the connection each computer will take up. Some routers allow you to specify the bandwidth available to each computer but you will have to check.
As for security issues, I have to point out that the likely hood of someone connecting to your network is very unlikely in domestic surroundings, and most people wouldn't have the know how to do it. However there is WEP which is a basic form of encryption and there are other things that can be done such as MAC filtering.