Your title suggests you want SPI, your question is about serial, which do you actually need?
A Mega has 4 serial ports, yes. The first of those 4 ports is taken up by the serial monitor, so is not really available unless you really know what you are doing. The other 3 are free for you to use as you please.
There is no problem reading and writing from / to all the serial ports concurrently, serial is a LOT slower than the processor, it will be able to keep up. The only thing is you must write cooperative, non-blocking code, which many people new to writing code struggle to get their heads around. Non blocking means no delay(), no while() and only very short and quick for loops, if any at all.