How do I connect a PIC microcontroller to a computer?
Progressing from the Arduino to a real system like a PIC requires more up-front investmant. Primarily you need to invest in a programmer.

Many development boards come with a programmer built in, but in general it is very handy to have one to hand regardless.

You can buy cheap Chinese cloned PICkit2 programmers on eBay, but I have found that the USB connectors on the ones I have used have been dodgy and needed replacing. The "normal" budget programmer these days is the PICkit 3 which deals with all the modern PIC chips, like the PIC32MZ series. For high end work and debugging you might end up wanting to invest in a better programmer, like a RealICE or something similar, though they are expensive. They give you much better debugging though, such as external trigger breakpoints which you don't get from a budget programmer.

A good transition from the Arduino's AVR chip to the PIC chips though is to maybe look at the chipKIT system - especially if you want to jump straight to the PIC32 chips. This is an Arduino compatible system based around the PIC32MX and PIC32MZ chips. Programming is the same as Arduino initially, but you can branch out and write raw PIC32 code if you so wish and program through the bootloader instead of needing a hardware programmer. It's still handy to have a hardware programmer to hand though...

For 8 bit PIC there are other boards available, such as the CHIPINO, but again that requires an external programmer to work with it.   A friend of mine also makes the Firewing which comes in a variety of forms, including a PIC18 version which has a bootloader and USB interface. It's designed to work with his own version of Basic, but there is nothing to stop you using MPLAB-X and XC8 to program it.

One caveat with PIC chips is that it can be tricky to work with a bootloader on them. You have to know how to work with linker scripts in order to fit in with the bootloader. Again, it is handy to have a hardware programmer to hand to reinstall the bootloader in case you manage to break it.

So in short: buy a hardware programmer, such as the PICkit3.