In short yes, you can run your own programs just as in other OS. It's probably easier in Linux than Windoze because the editor and compilers are generaly free too and in most cases included when you install the OS. What you can't do is write a program in say Visual C++ and expect it to run in Linux because some of the commands are specific to Windows drivers and DLLs. The underlying program structure is still the same though and there are equivalent Linux libraries that perform the same functions as the Wndows ones.
It's worth noting that Visual Studio programs can't produce or debug Linux executable files but there are several Linux compilers that produce code that works under both OS.
Please don't think of Linux as a replacement for Windows. Linux is an open-source version of UNIX which was multi-tasking and multi-user almost 20 years before the first version of Windows was released.
Why don't you download a "live" version of Linux and see for yourself. There are many versions which will run at reduced speed from CD/DVD or a USB stick so you can try it out without installing on your hard disk. I would recommend Ubuntu or Mint to get a feel of how it works. When you remove the CD/DVD or USB stick your computer will be completely unaware it ever ran the program and none of your existing files will have been changed. Linux can read Windows files natively so if you want to try out some of your programming, boot a live version of Linux, insert a USB stick and copy your files to it (so your originals are not touched) then try some Linux compilers.
Brian.