Can SPI and serial.begin be used at the same time? I have two modules where I want to control them through a single Arduino UNO.

I guess Serial.begin(9600) uses serial port and UART. But Im not sure about SPI.

Can I use SPI library hence the SPI communication together with serial communication(serial.begin) at the same time to communicate with two different modules?
Yes. UART and SPI are completely different things.  One uses pins 0/1, the other uses pins 11/12/13, and they use separate hardware and control registers inside the chip.