Ubuntu isn't usually installed on a FAT32 paritition. It uses either the EXT2 or EXT3 filesystem. It also needs a SWAP partition (this can be created upon installation with no real trouble. It only needs to be small. Mine is about 3GB but that's quite large for a swap partition).
It can read a number of different filesystem formats, including but not limited to FAT16, FAT32 and NTFS.
Nowadays, Ubuntu doesn't need anything extra installed to enable read/write support for NTFS partitions. It can easily automount NTFS partitions, and indeed, when you plug in any external HDD with an NTFS parition, these are automatically mounted.
It is quite easy to automatically mount any NTFS parititions on the same HDD as well, so you don't have to repetatively use the command line. There are plenty of tutorials on how to do this. The only problem that can arise with mounting these paritions is if windows wasn't shut down properly, so the NTFS partition is still in a "being used" state. However, Ubuntu lets you force the mounting process of these partitions, which will reset the usage state.
Until recently NTFS writing wasn't deemed "safe". However, since Ubuntu including write support natively, I have never had any problems reading or writing to an NTFS partition (i.e. nothing has been corrupted etc.)
So basically, No Ubuntu doesn't use FAT32, yes it can read/write to FAT32, and yes it can read/write to NTFS. In fact, Linux in general has better multi-filesystem support than Windows.