I am moving to a new place at the moment and will have a new laptop on which I would like to use my current savegame. Is there something I should know before doing that so that to make sure everything works 100%? Perhaps there are some tools to make a clean savegame or simply creating a new save will be enough to load it from a new computer?

Just Copy/Paste your save directory

You will also need to make sure that you have the same mod files installed on the new system (or else you will be missing the mod items and any mods that are mastered to other .esms will cause the save to crash if they are missing - so be sure you install any mods that you hae from the old system as well.

Easiest method would be to use a thumbdrive to copy the full My documents\My Games\Oblivion and Oblivion\Data folders from the old system to the new one after installing the game on the new system (installing adds the registry entries and then pasting the old folders will bring all of the saves and game files to the new system so you have an identical installation)