How do I play at the beginning of the game (spoiler warning)?

Solution 1:

Unlike FO3 and 4, the game has very few level scaling enemies - most areas have the same enemies whether you fight them at level 1 or level 30. If you run into an enemy or group of enemies you can't deal with, it is somewhat assumed you'll run away and come back later (experienced players may use specific items and tricks to defeat or sneak around them).

Something else to keep in mind is that many of the quests in the game, including the one mentioned in the OP, have non-violent solutions. I won't spoil it for you on that quest, but if you want to know what options you have besides fighting the Legion you can always look up the quest online.

With all that said, the beginning of the game can definitely feel tough, since it's easy to go a little way in the wrong direction and run into an enemy that is too difficult for you to defeat. In general you can try to stick to the main quest until you reach New Vegas, if you want - the locations it sends you to will all be low level, and I believe all of the quests can be completed without fighting. On the other hand, most of the side quests you are given should be intended for low level players, so if you are running into enemies that are too difficult, check your map and see if there are alternative routes to the quest location.

Solution 2:

A few tips:

1. Follow the road, it is always safer on the road than off it. At places where it is too tough for you, staying on the road can help avoid many encounters which will otherwise kill you. If you managed to reach The Strip safely there will be a special encounter that removes your negative relation with the Ceasar and they will stop hunting you until you go against them again.
2. Get a follower, where combat is concerned, a follower always helps more than any piece of equipment or weapon. By your description you should have reached Novac, there is one Companion there named Craig Boone which you can recruit to help you survive. If you have went on from Novac you can go further to 188 Trading Post where Veronica Santangelo is and recruit her instead.
3. Get stronger, roll back to before you enter Nipton, had you followed the road you'd have turned left at a crossroad before reaching Nipton, follow the road instead and you will arrive at Mojave Outpost. There are some quest you can do there. You could also backtrack and clear the side area along the way before Nipton for more exp and better equips before going to Nipton.
   A couple of place you can consider includes California Sunset Drive-in which have some lower level Rad Scorpions for you to practise against before you encounter higher level ones and Powder Ganger camp west which is between GoodSprings and Primm, you can get some explosives there which will help you in tougher fights.
4. Use VATS, this changes converts your ability to aim into a chance to hit, some people are better at shooting distance target with a scoped weapon while others are better at close quarter combat with a shotgun. Regardless VATS will have a fair chance of hitting a specific part of the body depending on your stats. But overall just use a suitable weapon for that combat range and you should have a fair chance of hitting.
5. Aim to cripple instead of killing your opponents. Against strong enemies it may take quite a while to killing them off completely, crippling the legs of enemies not capable of range attack will make them less threatening since they take longer to reach you so you can easily run away from them while taking down other enemies. Crippling the arms of enemies with guns will make their accuracy poorer, reducing the damage you take for the rest of the fight.
6. Maintain your weapon and armour, equipment below certain condition level will drop in damage or DT. That will directly affect your ability to fight.

Solution 3:

Fallout New Vegas uses enemies to "fence" you in.

From the start, if you simply walked north, towards your main story marker that points you to New Vegas, you will first encounter rad scorpions, then golden geckos, then deathclaws. None of these are beginner monsters.

Your focus really is to talk to everyone at the start, start doing some quests, and you will then discover level-appropriate locations.

If you go for New Vegas right away, you are basically expected to die.