Over the coming years, you will meet many people like those you met during your last day on that floor. There will be some great people to work with, both as co-workers and as patients and there will be some really horrible people too. The thing to remember is that you need to know what you're comfortable with, what you're willing to tolerate, and what you're not going to tolerate at all. When you have good, firm boundaries, you'll be amazed how quickly the toxic people stop trying to bother you because they realize that they can't bother you.
As to the IV starts, there will also be people that just don't tolerate pain at all and will scream, wiggle, shout, and the like from something that is usually a relatively minimal painful thing. It's a natural reaction to withdraw from pain. It's a self-preservation thing. However we can learn to tolerate certain amounts of it or tolerate pain when we know it is for a short period of time or for a specific purpose.
Don't get too enamored of the idea that ICU nurses are fantastic at doing things like IV starts. I'm an ED RN. I do IV starts (usually) several times a day on multiple patients and I used to be an active Paramedic. I've done this particular task for years. I really am that good. There are times that even I can't get a line. At my facility, I'm also ultrasound PIV trained and I usually can get those in one stick. It's very much true that even the most experienced of us can have trouble getting a line. I had a patient the other night that despite every trick in the book (and some that probably aren't), I just couldn't get IV access.
You were, however, given a great piece of education on your last day. You were presented a challenge that you felt you couldn't meet. You successfully said that you aren't comfortable with doing that particular task. It's often a difficult thing to say "no" (and even do it gracefully) when you're in a student/orientee/junior role. That's a huge thing to learn that you can do it.
Now I will say that when you have a chemo patient that has an implanted port, it is a clue that they're a difficult stick but they're not always a difficult stick. With port patients, I prefer NOT to access the port unless I absolutely MUST. There's just much that can go wrong and infection is just for starters. I've placed many a peripheral line in these patients. The key thing is to look and look, and look, and look, and look before you commit to doing the puncture. When I'm doing USGPIV's, I tell my patients that I will look and look and will NOT poke them unless I see something that I feel confident about. Same goes for regular IV lines. I had a difficult stick patient a couple nights ago and looked around a LOT until I found a vein that would likely work the first time. It was small but viable. I got it in one stick. I'm also not "married" to a given IV catheter size. I'll reach for an 18g or a 24g or anything in-between if it suits the need. These days I most commonly place 20g and 22g lines but I also happily place 18g as the need arises. Don't worry: you'll find your stride in this in due time.
Seriously, the biggest lesson often ends up learning to say "no."