The previous answers covered everything you need to know about money and stuff but I have some general survival tips if you're dying a lot.
1) Find a fiend to play with. This is number one for me because it's always funner with a friend and there's less pressure on you to kill EVERYTHING (although the more players there are, the more enemies spawn, so pick someone trigger happy). Being able to rely on a friend to heal you if you're fighting for you life and there are no enemies around to kill is priceless. Plus if you go down and they survive long enough for you to completely respawn, the progress is not lost.
2) You mentioned elementals. How much do you know about them? When I started playing, I got bored during Marcus' mission when he explained them all to me and I was just having fun messing up the psycho he had tied up to the target. I then learnt more about them and they've been a lifeline ever since. Here are the rules:
Fire - very good against flesh
Shock - very good against shields, moderate against flesh
Corrosive - very good against metal (loaders etc), moderate against flesh
Explosive - good all rounder
Slag - doesn't do anything on its own but hugely increases the damage of any elemental effect, including those induced by grenades or your skill tree
3) Now let's talk about weapons. If you're not at full ammo capacity, ALWAYS loot the crates you see about, even the tiny ones. It's the worst not monitoring your ammo consumption and having to switch guns if your main one is a legendary that eats through enemies like acid and you ran out of its ammo type and have to switch to a lesser weapon. When it comes to types, I always equip one of each of the following in my hot bar (but you can switch it up based on preference):
Pistol - good close to medium range, generally quite accurate
SMG - good close to medium range, fast rate of fire (good when you're panicking)
Sniper Rifle - good long range, accurate and easy to get a head shot
Rocket Launcher - medium to long range, great if you're facing a lot of or overpowered enemies, but can put you straight in fight for your life mode if you shoot something in front of you and you get in the way - this just takes a bit of practice and good timing
I also ensure my guns are different elemental types so that I'm equipped for anything (but if you know an area will be full of loaders for example, you can have multiple types of gun that are all corrosive equipped ready). I go for fire pistol, shock SMG, corrosive sniper rifle (allows crits on the loaders eyes and does corrosive at the same time), explosive rocket launcher. Sometimes I switch my sniper out for a slag one.
4) Use phase lock SPARINGLY. I play as Gaige, whose skill is Deathtrap, a robot killing machine, so I just spam F to respawn him whenever I can. However, phase lock is enemy specific (until you level up and learn the skill that allows multiple enemies to be locked at once), so only target badass or more technically challenging enemies with this and focus instead upon using the right elemental type for the enemy and being strategic in your attacks (pick off weaker enemies first so your attention is fully committed to the stronger ones when you reach them). Also grenades are incredibly useful when you have multiple enemies, just chuck one and run away to recuperate for a minute (health regeneration is also great so if you have a skill that allows this, get it!!).
5) Strafe!! When you're running to a destination and just want to get somewhere because the enemies are too overpowered or you can't be bothered to fight them, take on a diagonal path. If you were running away from gun fire in real life you wouldn't want to give them an easy target by being predictable!
6) Look into getting a shield upgrade which increases your maximum health. I can't remember off the top of my head what make those shields are (I wanna say Pangolin but not sure), but it gives you a bit more wriggle room if you're dying a lot.
That's it for now, but if you have any questions, let me know!