I went on a very long journey on default map generation settings to warmer biomes (and also to gather rare minerals like limestone and halite and any metal I found). Traveled roughly 50,000 blocks over some five months in winter (nothing to do at home). My strategy was to bring a cooking pot, bowl, and two stacks of turnips. Then, hunt meat or fish to supplement the veggies to make meals with. This worked fairly well while I was still in colder biomes. Fish was far easier as meat was hard to come by; I could mostly only find rabbits around, and those are harder to hit with spears.
Thing is, once I got far south enough that it was warm (or if I'd been traveling in summer) there was more than enough forageable food around. Berries, grains, termites eventually. I tossed out the pot, bowl, and just ate half the turnips raw to open up inventory slots. Takes less time than hunting/fishing/cooking by far. So long as you keep moving so an area doesn't get over-harvested, the land will provide everything you need as there is no detriment in the game currently to eating nothing but berries or whatever.
