"How do I start playing Mobile Frame Zero?" As with many other situations, the answer is, "It depends."
Back in September of 2015 I did another post about this issue. This is a different take.
Just want a squad for yourself? Find the instructions/parts list for the Frame you want to build, buy those parts off Bricklink/Brickowl/Pick-A-Brick, assemble your squad, find players, play game.
Intent on starting a MFZ gaming group by yourself and providing squads for all the players? Find sets that have the parts you want/need at a price you're happy with, build Frames, build Terrain, stockpile leftover parts, find players, play game, Rinse, Repeat.
Some where in between those points? Pick and choose from both methods.
A specific thing people might want to look at is the Double Set Challenge. In that I combine two small, cheap sets to make a Frame and possibly a Station. I've also been breaking down the costs to make Defender/Attacker/Primary Attacker size squads from those sets.
Storage? Tackle/tool boxes are good for storing and transporting squads or small collections. Have a larger collection that you want to organize in some fashion? Read this.