I start OA about level 5 or 6. I store any extra gear in the Nuka Cola machine in the subway station, keeping a rifle, that's it. Trust me, you'll have plenty to haul back when this is done.
I start the Pitt about level 15 or up. Again, barely need to bring anything, I bring a scoped magnum, picking up what I find. After you finish the Pitt, you will be able to freely go back and forth between there and the wasteland.
PL is reserved for the mid 20s, though I have gone straight there before. You'll want to bring some gear with you on this one. Flamer for sure, bottlecap mines for definitely sure. You'll have safe storage there, in a couple places. But, and no spoilers here, I would make some strategic saves throughout the main quest in PL, because you might have to reload when you lose stuff too valuable to blow off.
I don't have MZ.
There is a mission that you do, before Broken Steel gets loaded, where I go to the battle in my underwear. With 100% repair usually by this time, I pick up all the good armor and weapons along the way, repairing as I go. Easy way to net about 20,000 caps worth of gear. Though I don't do this anymore. Good luck. Save often, that's all I can say.