Now that you've successfully installed the core, you can move on to connecting your RP2040 to the Arduino IDE.
To connect your RP2040 microcontroller-based board, connect it to your computer via a known good USB power+data cable. Hold down the BOOTSEL button when you're plugging it in to enter the bootloader. It should then show up as a USB drive with the name RPI-RP2 (or something similar) in your computer File Explorer / Finder (depends on operating system).
You only need manually to enter the bootloader the first time you load an Arduino sketch onto your Pico. It is not necessary to manually enter the bootloader to load subsequent sketches once you are already running an Arduino sketch.
Then in the Arduino IDE, go to Tools -> Board -> Raspberry Pi RP2040 Boards and select the board you are using.
Now, you're going to want to select the correct port to use. Open Tools -> Port, and select the right port. On my computer, it was /dev/ttyS0. If it only gives you the options to use a port with ACM (Linux) in it, as in /dev/ttyACM0 or usbmodem (Mac/OSX), as in /dev/tty.usbmodem14301 then try unplugging it and plugging it back in, making sure to hold down the BOOTSEL button as you do so. On Windows, serial ports show up as COM ports.