Split it.
Go to Core City mostly right after the world opens up, and do the Arena fights until you have a hard time with one. Maybe you never do and run straight through; maybe you have to stop. Whatever. Hopefully you went ahead and aggroed your fellow challengers and took their stuff when you were fighting the groups; free loot.
Then go ahead and find Gorsky and do at least part of his questline. Poke around in the Drop Zone, and the sewers/undercity, hitting up those little fights and loot caches.
Then, having done the easy stuff in Core City, picked up maybe a level or two, and gotten a little bit of nicer loot, you can safely head over to Foundry and walk on through the whole thing.