How to learn dsPIC UART program based on PIC USART experience?
You have unfortunately not learned much from your previous UART experience if that is all the code you have. Here it seems like you have used a couple of libraries (i.e header files) which has implemented all the "tricky" stuff for you. 

I would suggest that you take a look in your code and open up the UART1_Init() function. This is most likely included as uart.c and uart.h, where the .c-file is the actual implementation. 

Here you will find how UART actually works, and when you have learned this you can move on to other controllers. As a matter of fact, look in to all of your uart-functions and study them, finding out what every line means. This is the most common way to learn code: To study code, understanding it and write your own.