What kind of partitioning should I chose?
Here are your answers:
- Yes, this is a correct partitioning.
- Ext4 is a default file system and it is optimal for desktop usage.
- If you do not have Windows installed, you do not need to create any Windows partitions.
- Fat32 partition will not make system slower, but I see no reason to create one at all.
Reformatting slightly, you wrote:
- 20GB - ext4 - / - for the operating system
- 4GB - swap
- The rest of 220 GB on 2 equal partitions of ext4.
Your root (/) and swap selections seem reasonable, with the caveat that your swap space should be at least as large as your RAM if you expect to be able to use a suspend-to-disk operation; thus, 4GiB (but maybe not 4GB -- see here if you don't understand the distinction) should be barely enough if you've got 4GiB of RAM. (Note that RAM is usually mis-specified using SI suffixes, like GB, although it's almost always installed in units better measured by IEEE-1541 suffixes, like GiB.)
The reasons you gave for creating two additional partitions (namely, better performance) are unlikely to be valid. Furthermore, the mount points are extremely important for this. I'd recommend putting the rest of your disk space in a single partition and mounting it at /home. An exception would be if you intend to use the system in an unusual way for a laptop. For instance, a server might want a big /var or /srv partition and no separate /home partition. There are reasons to split off various other partitions, but for an average new user, a single big /home partition is fine.
As to filesystems, ext4fs is the default and is perfectly reasonable. Other filesystems have their advantages, but optimizing your filesystem choice requires understanding details of your expected use that you haven't provided and that I suspect you don't even know. I'd stick with ext4fs.
There are few reasons to create a FAT32 partition on a computer that boots nothing but Ubuntu. Even if you copy files to the laptop that originated on Windows machines, using FAT32 to store them will provide no advantage. FAT32 has several drawbacks, too, including a 4GiB file-size limit, slower performance than most Linux-native filesystems, and lack of Linux permissions and ownership support.
The biggest exception to the FAT32 rule is if your computer uses the Extensible Firmware Interface (EFI) firmware (or its Unified EFI variant, aka EFI 2.x). Such computers use a FAT32 EFI System Partition (ESP) to store the boot loader. Most EFI-based computers can be booted in BIOS mode, though, so even on an EFI-based computer, creating an ESP might not be required. In your case, the link you provided to your computer model suggests that it originally shipped with Windows Vista, which means it's old enough that it almost certainly uses a BIOS, not an EFI, so chances are this doesn't apply to you. I'm mentioning it only in the interest of completeness, and in case somebody else who does have an EFI-based computer reads this post.