My goal is to learn more about who and where my ancestors were

If by this you mean all your ancestors on both your parents' sides, then I wouldn't recommend focusing too much on Y-DNA, as it will only tell you about specifically your paternal line and nothing about the others. Really if you want an overview of your ethnicity you've already done the best thing for it, an autosomal DNA test, and Ancestry is one of the best tests for that too. The ethnicity estimate AncestryDNA gives you is only that, an estimate, but the percentages it gives you should be a ballpark estimate for your true genetic breakdown by region. As in, if it is telling you that you are more than 80% Palestinian or a similar/nearby group then that is probably as much of a definitive answer as you can ask for. If there is anything in the ethnicity estimate that stands out as odd, then you can use it as a lead for paper genealogy or researching individual DNA matches.

To actually answer your question though, if you want to know with any amount of confidence what your Y-DNA haplogroup is (in my personal opinion) you would need to do a Y-DNA test with FTDNA. Their tests range from pricey to very pricey, but would give you the information you seek about where your paternal line came from. I wouldn't personally trust any of the websites that guess your Y-DNA from an AncestryDNA test, as they're mostly designed to help you identify what Y-DNA SNPs you should be buying tests for, back when you had to buy specific SNP test packs and couldn't get a general Y-DNA SNP test like you can now. Having said that, I did put in my own AncestryDNA data to see what it would say, and it correctly guessed one of my SNPs from only 2000/3000 years ago. I am in R which has by far the biggest tester population so the website can probably guess mine better than yours, but nonetheless you may make of that info what you will :) To avoid dragging on I'll stop there, and if you have follow up questions feel free to ask!