You have to explore the game as much as possible, look in all trash containers, tool boxes, etc.. Go into buildings that aren't marked on your map and clear them out. Check every corpse. Really, it isn't in the spirit of the story, but to be good at the game you have to be greedy. Leave no stone unturned. Gore bags tend to pay out the best treasure, IMHO.
With locked doors and boxes use the save feature to your advantage. Save game before you attempt to pick or hack a lock and if you fail, reload the game and try again. This will save you pins and also keep you from screwing the lock and not being able to access whatever you were trying to get behind it.
I disagree with collecting armour for cash. The issue is that it is very heavy early on in the game when you probably can't carry as much and you need more money than later on (when you'll eventually have piles of cash you can't even spend). It is easier to collect lots of semi-valuable lightweight items and leave the heavy stuff behind (if only temporarily).
Once you have completed the Megaton quests, supposing you do them in such a way that you gain a locker in Megaton, you can really start hauling stuff back and forth and using the base to stockpile items once the shops run out of money. Only pick up heavy weapons and armour for the repair parts and break them down as you need them at first. One trick is to collect everything in one location and stuff it into a fridge or drop it by the door or something, that way you can go back and forth from there to Megaton and totally wipe clean a part of the map (such as the Mirelurk sewer side quest). Use the quick travel feature your map provides as much as possible unless you want to spend extra hours walking to Megaton.
Without giving too much away you can pick up a female follower mid-way through the game when you are in the heart of DC looking for artifacts. Save your game frequently when you have a follower because they tend to rush into battle and get their head blown off. Don't take slaves if you want good karma.
Complete side quests before trying the main story quests, too. The more side quests you do, the stronger you'll get, and then the harder main quests won't be as bad. Plus you'll be better equipped.
Another tip is to stay well rested. You'll get more XP for kills and quests when you are rested.
