I've only been bitten by non venomous snakes myself--once by a python (it was about 10' long), which hurt like any animal bite would. It was a very quick thing. Its teeth were very small, but there were a lot in its mouth and they point backwards, so there's some tearing when it pulled away. There was bruising and my hand swelled up some, and I went to the doctor. They gave me antibiotics as a precaution (since their mouths have some interesting bacteria) and a tetanus shot.
I was also bitten by a large bull snake (one of our bio departments animals) when I was handling it in class. Mr. Big is usually docile, and his bite was pretty lackadaisical (I think he smelled another animal on my hand and thought it might be supper). He held on, the way he might with a mouse. I gently took his head and pushed it forward and off, so he released without tearing my skin, and I just had tiny pin pricks from his teeth. The wound wasn't deep and didn't hurt all that much. I cleaned it up with iodine. And I always wash my hands before handling that snake now.
There is a visceral startle reaction when something bites you, even for those of us who aren't afraid of snakes. And I think humans might be wired up to jump back when a snake appears suddenly near their foot. I've had benign snakes slither right in front of me while I was hiking, and I always jump back with heart pounding, even as my brain is registering, "Oh, that's just a coachwhip. Cool." or whatever.
I knew someone who was bitten on the toe by a baby rattle snake when he was canoeing. He said it felt like a bee sting initially, but there was the snake (and two tiny fang marks on his bare toe). He got very little venom with the bite, but like an idiot, didn't go to the urgent care for a couple hours, until it started to swell and he felt woozy. He ended up being okay, but yesh. We told him he was an idiot, because rattlesnake venom can cause gangrene and loss of digits (or limbs).
If you're making up a magical snake with magical venom, you can make its effects feel pretty much any way you want. Venom often burns, tingles, stings, or feels intensely cold when it enters someone's body. With very intense venoms, the victim can start to feel sick, dizzy, have palpitations, sweat, cramp, hallucinate, or lose consciousness within minutes or less of being bitten.
One thing--cutting the wound and sucking the venom out of a wound by mouth or suction bulb (as per outdated "snake bite kits" and an old standby rememdy in many a novel) doesn't actually work, and it can actually do more harm. And compression bandages are a terrible idea for hemolytic toxins like rattlesnake venom (though I understand they're the first-aid treatment of choice for some kinds of venoms, like funnel web spiders).