Ouch - small and hard list of quests for a beginner. My typical start game involves:
1) Tutorial. Yeah I always do this even if it doesn't give me much.
2) Primm. See if you can find a companion. Do the sheriff quest and fought the law quest unless you do it on your own.
3) Outland station south of Primm (where you picked up Heartache by the number). Has a couple of easy quests...
4) ...part of which is going to Nipton. Continue to Novac to get Boone and do the other quests here. Of which one is outright awesome.
5) Or pick up Veronica at the trading post. Do any quests you find enroute to Boulder to complete your "They went that away".
After "They went that away", then my character is usually well enough developed to really start the game. Do I want to focus on guns and speech? Then those skills are pretty high by now, maybe coupled with a third utility skill (often medicine or repair for me). Explore the wasteland. Wreck havoc in fiend territory. Map out locations even if you find you're too weak to enter them. Stay away from quarries for now

And if you see fast moving red dots, try to avoid those too. Pay attention to you caliber - is it really working against you foe? Do you really need lots of armor, or is it more valuable to get other bonuses such as criticals, guns, and agility?
On my first playthrough, I got owned as well. Mostly due to own stupidity. "Warning - deathclaw ahead" - that's a game version of an invite, right? Wrong

Clearing out the NCRCF with a 9mm and non focused training *was* possible (non hardcoe, and without companion), but it was *not* easy. But you learn from your mistakes, and the second playthrough was pretty much barging in guns blazing, as I was more focused in training and had obtained better guns. Hard to believe my struggles during the first playthrough

The MQ "Ring-A-Ding-Ding" that follows "They went that away", you may want to hold off for as long as possible. When you get to the point where you can approach "a certain someone", avoid doing so, and instead explore as much as possible. Your quest box should fill up quickly. Take manual notes of so called unmarked quests.