The Brain0110, loading up a save from one computer to another should work, so long as the mod versions are ALL the same versions, and the mod SETTINGS are identical---if they aren't, there could be issues either with map generation, item/technology/recipes, etc.
To make it simple, factorio puts all mod files and mod settings in a single folder, depending on where you've installed the game, locate that folder and transport it to your other computer you'd like to play the game on. the modfiles.json contains a list of files in the folder and a true/false flag for if that mod is enabled, the other important file in the mods folder contains all config settings the mods have. there are a quite a few mods out that,if you change a config setting for a mod and try and load up a world, it can break lots of things (remove items from the game, mess up your research, all sorts of crap can happen). (although their filetype should be kept as json, they are basically nothing more than a text file, and you a text editor such as notepad++ is great for reading or editing stuff like that)
