It depends what you mean by changing your address, and what your network configuration is.
When you are browsing, the IP address that the outside Web sees you as coming from is the WAN IP address assigned to your broadband modem by your ISP (Cable or DSL) You cannot change this assignment of IP address. A dynamic address may last for several weeks or months but will change periodically as they reassign a number to you out of their pool. You may be able to trigger a change if you power-off your broadband modem. This WAN IP number will be part of their assigned block of addresses and you cannot control what you get. So your computer will still be identifiable as, for example, a Comcast cable customer from Philadelphia PA.
If you are talking about using your computer’s internal network settings to change its IP address, then it depends how your network is configured. In a typical network, the router or the broadband modem performs NAT (network address translation) and automatically assigns your computer an internal network address. This address is within a restricted range of non-public addresses reserved for internal network use, such as 192.168.0.xxx (different brands of routers have different default gateway addresses). If your computer is set to get the address from the router or modem by DHCP, then the router or modem automatically issues your machine this address.
You can set the network settings of your machine to enter your own IP address manually. If you do so, it must be within the range of the router’s internal addresses. So you could change from 192.168.0.101 to 192.168.0.150, and all would be OK. But if you changed to 175.1.0.101 or even 192.168.2.101, then your Internet would stop working because you would no longer be in the same subnet as your router.
But whatever you change your internal address to, makes no difference in the way the external Web sees your public IP address, which will still be the one that the ISP has assigned to your modem.
If you want to be anonymous by using a different public IP address, you will have to make use of a proxy, which is a way to ‘bounce’ your Web requests off of another, outside computer system. Before you go down this route, you should know that many services that require to know your IP address, like online streaming TV from Hulu, are well aware of proxy systems and continually block proxied addresses.