Hardcore for me involves a little extra planning.
I start off with a decent endurance (8 or so, all implants + nice boost to HP/poison resist etc). I prefer energy weapons, sure the ammo is more expensive, but you don't have to carry several kinds for your weapons, just some MF and SEC cells depending on what you are carrying.
The Survival skill is your best friend, make Desert Salads and Gecko Kebabs, and if you are in a pinch use Maize (lots in the Boomers area, and Sharecropper farms).
The Science skill lets you make Hydra (which, IMO, is WAY better than a doctor's bag) it weighs nothing and constantly restores your limbs. One hydra often heals my limbs to full from crippled. Materials needed are a bit hard to find until you go into Honest Hearts DLC, but Cave fungus can be found on Legion troops semi-often.
Barter skill gets you access to several perks that let you carry more stuff or reduce the weight of all items under 2 pounds, like "Pack Rat" and "Long Haul" If you have Old World Blues, try and get the perk "Them's Good Eatin'" for some uber healing items from just about everything you kill.
Followers die permanently in Hardcore, as was mentioned. Be sure to gear them well, and be careful of enemies you engage. Try to keep your distance, and remember not getting hit at all is better than trying to strafe a charging cazador / deathclaw.