Firstly, don't despair.  I'm of the opinion that the game broadly gets easier as you go along. There are some bumps along the way, but I spent more time getting out of the first area than any other.

I don't think there's a right way to play per se. I tend to use a lot of cover, and take it slow. The more you rush into a room the more likely you are to get swarmed by enemies, so I take my time, exploit the environment and pick things off at a distance, only rushing in to get the cash. My daughter on the other hand flies through rooms like a girl possessed and melees absolutely bloody everything. She has more money, I have more health. Still, it's worth experimenting a little with your play style, to see what works.
Don't forget to check your map.  Not just for goodies to pick up, but also for enemies you might not have spotted. I never left a room without mopping up absolutely everything inside.  (Unless you want to try Crayon's trick from the previous page.)
Similarly always explore.  Use the tentacles as lifts, look for walls that can be destroyed, and blow up statues with glowing eyes.  
Save keys for doors, rather than chests.

I always threw ether into the machine at the start. That way you unlock a new toy for the run, and the option of it appearing again later. Some are very much worth having and can absolutely change a playthrough. (Some less so). I wouldn't spend ether on anything else at this stage, except for regeneration just before the boss. 

I would generally avoid any malignant items early on, unless you have a nullification sphere. Certainly don't waste ether cleaning them.

And if you're avoiding malignant items you can happily scoop up any parasite that increases risk of malfunction, or severity of those malfunctions, without fear. (You'll get a feel for the different parasites, but the "survive a killing blow" one is never a bad decision, whilst the one that regenerates health when low is usually a life saver too - especially when facing bosses.)

If you have the parasite that causes damage from a large fall, remember that dashing just before you hit the ground will save you. The catch being that I almost always forget.

Don't be afraid of the Challenge Room. I would usually save it for last, but it's worth doing.

If you get a chance to buy a spaceman, then do.

Practice the Overload. I still suck at it, usually because I forget it's there. But it can make a huge difference in difficult battles - so practice it in the easy ones.

If you're being swarmed by enemies, with nowhere to run, focus on whoever's about to attack first (several have tells) and try to stagger them. The minute they're staggered (they'll drop the attack sequence) move onto the next one. This is particularly true with any airborne enemies.

Don't forget your melee. Some of the bigger beasts are very susceptible to it. (The shielded one that spits fire, and the tall tree like Groot monsters, both struggle to get up during repeated melee.)

Most of the time it's perfectly possible to flee a room. A small number of enemies can follow you (those Groot things again), but it none the less helps to balance the numbers. (Though I'd caution hiding behind doors too much.  If you can't see them, it's hard to dodge their attacks when you open the door again.)

Finally, learn the rooms. There are only a limited number that rotate in. I've seen this criticised by some, but in reality what it means is that you have a series of arenas that you can learn, and by extension get better at exploiting. (It's the other advantage of getting all the prizes before you leave, scooping all that stuff up helps you find all the nooks and crannies, and orientate you in future runs.)