You should try to get the right skills to level up faster:

1. Max out your Intelligence to get more EXP per level (Alternatively you could upgrade your luck to 5 and the skill Idiot Savant to get randomly more EXP the lesser you Inteligence is. But Intelligence has to offer some usefull skills, so I wouldn't make it that way.)
2. Upgrade the skills Locksmith and / or Pickpocket (you'll need Perception), Hacker (you'll need Intelligence which should be at 10 because of 1.) and Charisma to persuade the people. Each successful event will give you EXP (and some loot). You don't need level 4 of Locksmith and Hacker.
3. While this will take some levels (regarding your starting skilling) you should explore every point of interest on the map. On your way you can use the skills from 2. and kill some monsters.
4. Also try to collect many materials for building weapon or armor improvements (therefore you could also increase the skills Armorer and Blacksmith (level 4 and 5 of strength), Gun Nut and Science! (level 3 and 6 of Intelligence)
5. Thanks to nickson104: Build things in your settlements. Collect Ressources to use them at the workbench.
6. And of course: finish as many side and main quests as possible.

Hopefully I mentioned everything. You could give this side a try where I already have choosen a good basic skilling for your case.

Killing big monsters isn't the thing. They are really hard and don't give you the EXP which is worth your ammo and / or stimpacks.

Good luck, wastelander!

An Idiot's guide from Level 1 to Level 11+

If you want to somewhat spoil your game experience, you can gain a lot of levels fairly quickly by exploiting the settlement building feature of the game.

I'm going to write this post under the assumption that the majority of people who come here will have only just started the game or are wanting to do this on a new file. In your case, where you've already started, the below strategy is still an option but it won't be as profitable if you don't already have the right build.

Step 1. Starting the game with the right build

You'll want the following stats:

1. No more than 1 Intelligence.
2. A minimum of 5 Luck.

With that out of the way, you'll want to go out and get yourself to Level 2. Do this by speaking to Codswold and completing his initial quest. With this quest out of the way, you'll now be able to access Sanctuary's Workshop menu.

Being level 2 will give you the ability to unlock some of the Perks the game has to offer. The one we want to unlock first is under the Luck attribute's column and is called Idiot Savant. Getting this to its first level will mean any experience-giving action you make will now have a chance of having its experience tripled.

The reason we kept our Intelligence stat at 1 is because until you have a lot of Intelligence, this reduces the effect of the Idiot Savant perk.

Idiot Savant

Rank 1 · You're not stupid! Just... different. Randomly receive 3x XP from any action, and the lower your Intelligence, the greater the chance.

Rank 2 (Level 11) · You now randomly receive 5x XP from any action. The lower your Intelligence, the greater the chance.

Step 2. Scrap, scrap, scrap

The next thing we want to do is run around Sanctuary converting everything into Scrap. You can do this by opening the Workshop menu and running up to an item and executing the Scrap action.

Step 3. Build, build, build

Now that we've gathered a lot of resources, it's time to build. Anything you build will yield experience, and the Idiot Savant Perk we unlocked above will triple some of this experience. Pick an item of furniture and place it. Then place it again. Place it some more. Keep placing it. Place it anywhere you want.

This step is pretty creative; why not try building yourself a stairway to heaven with rugs? Remember not to die on the way back down though...

Step 4. Scrap, scrap, scrap

Now that we've build a load of stuff, it's time to scrap it all. You'll get half the resources required to build the item back. Not bad, huh?

Not high enough level? Go back to Step 2/3 and repeat the process until you're satisfied.

At level 11 you'll want to unlock the second level of Idiot Savant to increase this 3x multiplier into a 5x multiplier.

An Idiot's guide from Level 11 Onwards...

Now that you're Level 11, things are getting a bit tedious. It's time to stop building and start exploiting quests! From now on, all you'll need to do is save before you complete a quest and keep reloading until your shiny 5x experience multiplier kicks in. Why settle for 300 experience when you can have 1,500 experience after a couple of reloads?

And it's not just quests. Anything you do will receive this bonus. Picking a lock? Boom! 5x experience. Cracking a safe? Boom! 5x experience. The Commonwealth is your oyster.

• Raise your Intelligence. The higher it is the more XP you get.
• Idiot Savant perk. It will proc more the lower your Intelligence, but it still is useful even at Intelligence 10+.
• Sleep before turning in quests/settlement building. Being well rested gives you a bonus to XP.
• Have a Romantic companion. The Well Rested bonus is bigger if you sleep with your lover.
• Wear your smartest clothes. When settlement building or turning in quests put on your clothes/armor that has the highest Intelligence bonus.
• Take Mentats. Any drugs that increase your Intelligence increase your XP gains.