CrossDOS is an integrated part of AmigaOS/Workbench 3.0. Your A1200 should have OS 3.1 or 3.0 so you're safe.
If the Amiga you bought came with manuals, check for the manual covering "Workbench". I know it's boring to read manuals but if you've never used an Amiga before, this is actually a good start. Among other things, it tells you how to use CrossDOS.
Regarding the hard drive, the A1200 has an internal 2.5" IDE connector so if you can get your hands on an 2.5" drive (normally these are used in laptops) this would be a good start. It really doesn't need to be bigger than a couple of gigabytes so if you're lucky you can get an old PC laptop really cheap and just use that hard drive.
If the hard drive is bigger than 4GB you will have to use some third-party software to utilize the extra space unless you just decide to keep the total size of all the partitions below 4GB.
You can also use 3.5" IDE hard drives but then you need to do some (pretty simple) hardware modifications.
You can also pick up a few pointers here:
http://wonkity.com/~wblock/a1200/a1200.html