Which board did you buy? Most of them (Zedboard, Zybo, MicroZed) come with a Linux kernel already burned in QSPI flash. Make sure the jumpers are set for QSPI boot (not SD card). Plug in to the USB serial CDC port to see the prints. The system should boot to a login, password is usually root.
Now, download and install the Xilinx SDK. Write a hello world program. Run the settings64.sh script that was installed to get the toolchain in your path. Compile the code with arm-xilinx-linux-gnueabi-gcc. Put the output ELF file on a SD card. Insert the SD card to your board, mount it, chmod +x your exe, then execute it.
If your board didn't come with Linux, find a prebuilt kernel and put it on a SD card. There are usually just two files you need, BOOT.BIN and image.ub.
Try reading the Xilinx Petalinux documentation. It's quite good. Also try Sven's Zynq tutorials.
Finally, the Zynq has a complex bootloader architecture. If you want to run bare metal, your bare metal app can run in place of the SSBL (U-Boot). And you don't need to load the FPGA at first, the Zynq works fine if the PL is not configured