Please confirm below details.
10.5 is the latest version (released on 27th May 2021).
Recommended version is 10.2.1.
Now come to your question.
Can I update right to the newest update or do i need to do it in steps?
The answer is, That depends upon the current version. Release notes of a specific version will tell you exactly from which lowest version you can jump directly to it.
For example, if you wish to upgrade to 10.2.1 version and your current version is on at least 10.0.0 or higher version, then you can jump directly to 10.2.1.
This is nicely documented in each version's release notes under "Supported Upgrade Paths". In case of 10.2.1 release notes, it is in page number 5.
If that is the case, you can upgrade to 10.2.1.0.200. It has FragAttachs fix as well.
All the APs mentioned by you are supported and you can directly upgrade to this version from 10.x.