How do I learn arm programming

Once you put a heavyweight OS like Linux on a device, the level of abstraction from the hardware it provides makes it largely irrelevant what the chip is. If you want to learn something about ARM specifically, using Linux is a way of avoiding exactly that!

Morover the jump from PIC to ARM + Linux is huge. Linux does not get out of bed for less that 4Mb or RAM and considerably more non-volatile storage - and that is a bare minimum. ARM chips cover a broad spectrum, with low-end parts not even capable of supporting Linux. To make Linux worthwhile you need an ARM part with MMU support, which excludes a large range of ARM7 and Cortex-M parts.

There are plenty of smaller operating systems for ARM that will allow you to perform efficient (and hard real-time) scheduling and IPC with a very small footprint. They range form simple scheduling kernels such as FreeRTOS to more complete operating systems with standard device support and networking such as eCOS. Even if you use a simple scheduler, there are plenty of libraries available to support networking, filesystems, USB etc.

The answer to your question about compiler is almost certainly GCC - thet is the compiler Linux is built with. You will need a cross-compiler to build the kernel itself, but if you do have an ARM platform with sufficient resource, once you have Linux running on it, your target can host a compiler natively.

If you truly want to use Linux on ARM against all my advice, then the lowest cost, least effort approach to doing so is perhaps to use a Raspberry Pi. It is an ARM11 based board that runs Linux out of the box, is increasingly widely supported, and can be overclocked to 900MHz