The game scales to your level, so at the start it's pretty rough without having a good gun. As you level up, you will get better drops and do more damage.
This game does take time to find the best gun for you play style. Not all guns will damage certain enemies well. If the enemy attacks you with a specific type of damage, (ie poison, fire, etc) then your weapon that does that type of damage is not going to be as effective. You will need to carry a few different styles of guns for certain situations. Keep at it and as you level up your characters skill tree, they will get MUCH more powerful. You should do the missions in the order of easy first. If you are attempting a mission where it's HARD or IMPOSSIBLE you of course will not do any damage and it will be like going up against a wall. Come back to it later when it is listed as easy or trivial.
Read the mission details carefully to determine which mission to do and you will be fine. I tend to switch between the map and mission screen to see what I can 'finish' in the 'zone' I'm currently in, instead of backtracking all over the zones. This game is HUGE.
By the way.. after you 'finish' the game.. there will be a 'game+' mode where you can restart again with ALL your stats and guns carried over and let me tell you, THAT'S when this game shines.. the drops get a LOT better and the money too. If you get hooked on the game, you'll find yourself playing through the game with EACH character and most likely play through twice.
I suggest for your first playthru to use the soldier class since he is the easiest to use.
If you get stuck or want some help.. this link is a great resource. http://borderlands.wikia.com/wiki/Borderlands_Wiki