Puking is my ultimate weakness. There's nothing that I hate more. Not even getting kicked in the balls. I can only get half-drunk, the maximum I can get through is 3 bottles of beer and a mug of vodka in one night. When I puke, I NEVER continue drinking. I just start puking an puking until my stomach is completely empty.... and I still keep puking nothing!

I can get rid of the hangover really fast by just drinking a lot of water. But my upset stomach lasts for days. I tried training my body for months to tolerate alcohol, but no progress at all.
Hey don't drink more than you can handle, it takes time to get used to it and to understand your alcohol tolerance, and which drinks your stomach likes/which ones it doesn't.
I started 4 years ago and like you initially I could only have 2-3 and I'd get drunk and pukey. I think that's because it was all new to me so I wanted to try everything and join in with the others (drinking shots etc), and I learnt the hard way lol (like you, I hate puking!), e.g. I used to love tequilla and now I can't stand it, so you'll learn with more experience.
It is also good to make a habit of drinking water in between drinks, it'll keep you hydrated, help you pace yourself and keep the drink/food down.