Well it depends very strongly on which Alt start I pick. For example Talon and Outcasts start with pretty good armor and weapons right off the bat, so you can start terrorizing Raiders pretty early on. Regulator is also great since you start directly at the Regulator HQ and you can simply grab the Unique Lawdog pistol in there and start killing evil doers pretty effectively
For more regular starts I just scavenge and steal as much as I can. I think it's especially vital to get the parts for the Repair and Med Kits early on, since the bonuses are a big help when your skills are still low. Super Duper Market is always a good start, since there are a lot of corners to pop out, give off a few shots and duck back. Grenades and mines are also pretty essential. Next stop is usually springvale school to get my hands on an assault rifle and again mines are important here to keep raiders from flanking you. After that I usually have enough supplies to tackle other areas. Greyditch is also a great early stop, since the ants can easily be killed with hunting rifles and sawed off shotguns and you get some great rewards and loot there.
Great tips! You've given me some really good ideas for when I get my hands on some mines early for my next playthrough