Save your game often. The skill checks are much harder than in Fallout 1. Particularly before stealing, do save your game. Even with 10 dex, 300 stealing and sneaking skill, with the Pickpocket perk, and sneaking from the back (although it shouldn't matter with that perk), I did often botch stealing small stuff, like stimpacks.
Save before _exiting_ a combat or before finishing a quest, if you're close to level up (above level 4-5 or so), and have npc's in your party. NPC's have a chance to level up, too, when you do. Mostly, at level ups where you get a perk, I think, though occasionally it did happen to me at other levels, too. So save, and then exit combat. If you levelled up, and no npc did, reload and try again a few times. You don't want to carry around a bunch of wimps, when you could have almost decent fighters with you. Remember though, that npc's can only level up only so far, and only so often. If they've all had 5 level-ups or so, they can't get any further, so don't bother.
Steal lots. Some stuff, like the Bozar (or whatever... can't tell "a" from "o" on their funky font) I only saw in one shop, and even there only after doing a bunch of quests, and very expensive. On the other hand, you can get 3 of them by stealing from the green armoured guards outside NCR. Other stuff, like ammo and stimpacks, are just too expensive to buy in shops.
Be prepared for serious financial trouble. In Fallout 1, it was damn easy to swindle merchants. In the sequel, it isn't. I had around 200 barter skill on one character, and over 2000 karma, and was idolized in that city, and still my goods would be priced 2-3 times less than the merchants' goods. Plus, now most npc's leave very little cash and stuff to sell (particularly, most don't leave armours.) Briefly, you can't run around buying everything you see.
Watch your step. Particularly when you get a party, and give them ranged weapons, they just _love_ using burst mode and spraying bullets at an enemy that's _behind_ you. I had an adrenaline rush each time I saw Marcus pull out his minigun.
Watch your aim. It seems a lot easier to accidentally hit a team mate or inocent bystander, than it was in Fallout 1. Even with insanely high skills. Particularly if you use burst mode.
Avoid mixed encounters. If you encounter something like "a patrol fighting highwaymen", just say no, if you get the choice. If not, just run like hell for the exit grid. Unless you feel tough enough to fight _both_, that is. Usually, either a patroller will hit one of your men, or viceversa, and next thing you know, both sides are shooting at you.
When you get a new gun, don't just look at the gun's damage. Also examine the ammo. The ammo almost always has its own AP, DR and final damage modifiers. Yes, that's on top of the weapon's min-max damage numbers. Yes, these effects can go as far as making a weapon better than another, even though the min-max damage numbers say otherwise.
Forget about AP ammo. I think the game's model is plain screwed up. AP ammo does penetrate slightly more often, but does _insignificant_ damage, if you ask me. The only exception I know of is the needler ammo, whose AP ammo has a damage modifier that looks suspiciously like the JHP modifier for other weapons. Personally, I've found JHP ammo to be the best, even against the best armoured opponents, which is why I say the model is screwed up. JHP should realistically just go splat on armour, and even more so in the case of metal plate armour or power armour. For that matter, forget about the 14mm pistols, when you find them: I only found AP ammo for them.
Specialize. As I've said, some skill checks are insanely difficult. You can't bring all your skills to high enough levels, even with the "Skilled" trait and 10 int. So screw some skills. E.g., you probably don't need thrown weapons, since all grenades were very disappointing for my taste. I also screwed melee weapons, since you can get just as good weapons based on the "unarmed" skill, and you need unarmed for a couple of quests. Energy weapons can also safely be ignored, since by the time you get them, your enemies will be using "Advanced Power Armours" which make the Gatling Laser almost useless for the ammo weight used, and that's the toughest energy weapon I've found. For that matter, it IS possible to kick righteous donkey with just the "small guns" skill, and without the large ones, particularly with the "fast shot" trait. Your call, though.
Think in terms of action points, not in terms of what's natural. E.g., if you've opened your inventory in combat, make damn sure to do everything you need. You can reload/change two weapons with 120 bullets each AND use 10 stimpacks in a row for just 4 AP, or just 2 AP with the "Quick Pockets" perk. (Tho, personally, I'd say other perks are more useful to pick, but it's your call.)
For that reason, reload your weapons after every combat. The rest of the ammo in a weapon is not lost if you reload, and it helps if you don't have to waste time reloading at the beginning of the next fight.
For that reason, again, you may want to judge a weapon not only by the damage per shot, but also in terms of how many shots you can do per turn with it. Should be obvious, tho.
If you get a good tight group or line-up of enemies, go burst mode on them. If you've got one of your own men in the way, don't. Actually, if you got your own men in the way, don't go for an unlikely aimed shot, either. (E.g., don't for the eyes, in the dark, from far away with a short range pistol, when it says you have 17% chance to actually hit.) Chances are you won't just miss and hit another body part, as would realistically happen, but you'll often critically miss and hit someone else instead. Yet another game model screw up, if you ask me, but be warned nevertheless.
Check out areas obscured by walls. I.e., move close to the west and south edges of all rooms and corridors. The designers seem to have _loved_ placing containers hidden by walls, and there's good loot to be gained that way.
If you decide to have a weapon upgraded, get the ammo out of it first. When you get the upgraded weapon, what the game effectively does isn't upgrade your item, but _replace_ your item with another. And it'll be filled with new ammo. Hey, it may not mean many bullets, but it's for free. (Same applies for upgrading flamethrower fuel, btw. Fuel, like all ammo, is handled in packs, not by piece. By loading/unloading your flamer, you can create incomplete packs, like, say 5 fuel tanks, which upon upgrade will be replaced by a full 10 fuel tank pack. OK, so it's cheating :)