Usually you don't run into anyone wanting to fight you in a town unless you walk into hostile territory or trespass somehow. If you offend someone through speech (easy enough to do) you may end up in a fight, too. Where are you running into problems, specifically?

Put your weapon away (de-equip it) when entering settlements. After clearing the rats outside vault 13 and exiting the cave, travel directly to vault 15. You'll come across a friendly town about halfway there.

After killing all the rats at the beginning of the game, leave the map of Vault 13 and then immediately return. The door should work now so you can get some stuff there.

Depending on your stats and skills, you should be able to get some decent equipment there, while gaining some experience to better prepare you for the outside world.

Then do as they say above and get more equipment and experience at the friendly village.

When heading into a battle heavy area Save is your friend. it's easy to get killed when the computer gets a lucky critical. That will happen even late in the game.

Also HOARD STIMPACKS. They're really expensive early on, so it can be difficult. Also try to maximize their result by only using them when you're 20 below max health (until you build up a stock and can afford to use them less sparingly).