Can we clean out our water pipes

There is an acid that a plumber must pumped into the system and prevent the solution from getting the water supply. It works well BUT (as if you didn't know about "the but clause") the acid can also eat away at weakened joints in the pipe and spring leaks, which obviously adds to the cost. Best bet, Buy the new softener and filtration system, have the water tested and adjust the softener accordingly. The buildup in the pipes will eventually be cleansed out without any structural damage to the pipes. It's not going to happen overnight or in a week but it will happen. The manufacturer or the technician may set the initial backwash of the system to occur a couple of times a day and the reset to once a day. Ask questions before you buy and understand what the system will do regarding the pipe buildup.

Your best bet, if you are referring to your water pipes, would be to replace them with either copper or PEX. By saying you have "rusty pipes" tells me that you likely have galvanized water lines. You house probably has some age on it. Over time, galvanized, being the metal that it is, will rust and deteriorate and WILL cause you problems. By replacing the pipes, you will increase the volume of water flowing through them thus increasing your pressure. Good luck.