How to Get Studs
Make getting the multiplier pink bricks a priority- not only will the studs come rolling in, but they'll make hitting the True Believer challenge on every level a cinch. You don't actually need the last two multipliers, but they do more than pay for themselves over the course of the game.
The best place to casually farm purple studs is the entrance to the Sakaar Stadium. Destroy everything inside the entrance passsage, then the two passages left and one passage right. Do this every time you return to the main map to build up a tidy sum. Mission wise, the best bet is the High Noon Saloon (Yes, Eson has tons of studs, but it takes an annoyingly long time to get to them). In the Saloon, take a brawler like Kamala and start beating everyone up, especially civilians. Once your multiplier hits ten, gather up the blue studs in the periphery of the room, and particularly the purple stud stage left, which is now like getting ten purple studs.
Exit to Chronopolis once you've cleared the initial fight, then return to to your regularly scheduled questing, or stop by the Sakaar Arena for a little rinse and repeat.
Important Note on Multiplier Bonuses!
Purchasing one of the Pink Brick stud multipliers does not automatically activate that multiplier. Be sure to visit the 'extras' menu and toggle the multiplier on, otherwise you'll just be earning studs at the same old 1-to-1 ratio, like some kind of caveman.