So I decided to download a game onto a memory stick and its running fine
I named a profile so I could see if it worked on my sister's pc
Game runs, but the profile I made did not show up.
My sister made a profile, logged out, then back in. There's the one she made
So i go back to my pc, run the game, and there's my profile but not hers

How do I find out where the game is saving to so I can play my profile on any pc?

It's unclear if you are using Galaxy or you use offline installers and some in-game profile system. I don't know about Galaxy, but for games you can often find the save game/profile/configuration file location on PCGamingWiki website. There are no rules where the game stores that info; it can be in the registry, my documents, installation folder, in one of the user AppData folders or in ProgramData folder. It's a complete mess.

What is the game ?

Most modern PC games will store saves in a user specific area, sometimes the user folder, My Documents or AppData/ProgramData as antrad88 said. Where they're stored depends on what the dev chose, but they pretty much never save in the game's installation folder any more as that's considered bad practice.

Right now its SpaceChem and that site did help somewhat
I was trying to follow the path manually for when I try other games and I get dead ends. Like AppData does not show except when I search. Then I go a few more files in an the next folder does not show.

Oh yeah, AppData is a hidden folder, you won't see it normally. I don't know how it works on Windows 10, but in Windows 7, you need to press Alt to display the menu bar in any open window, then click Tools->Folder Options->View then in advanced settings check "Display hidden files and stuff". Then you can click "Apply to other folders". It is not recommended to leave this enabled unless you know what you are doing.

Alternately, opening the run window (Windows button +R) and typing %appdata% will open the appdata folder even if it's hidden.