There are also different color peppercorns. Green is unripe and usually dehydrated giving a slightly herbal taste IMO. White is normal peppercorn but the black part is removed giving a hotter taste IMO. White is preferred mostly for the color not the taste however. Pink is not actually pepper it just looks similar. As for black peppercorn, Tellicherry are actually just larger peppercorns that are sometimes hand picked, inspected, etc. I have not noticed Tellicherry having a different taste than another black assuming it is from the same region at the same time (same pepper just bigger corns). Black will vary a lot in taste (within the pepper scope) because most you buy in a store you have no idea where it came from. There's a lot of misinformation on black pepper, but it mostly all comes from India.
My wife and I just did some peppercorn taste testing as I was experimenting with my brisket rub, naturally she preferred the Sam's Club big shaker black peppercorns from a few years ago. It was definitely what we both associated with black pepper taste being what you get most places over our middle aged lives... We did agree on the tastes I mentioned which is what I was expecting from previous experience.
The first step I would take is fresh cracked. Then try the different peppercorn colors. Beyond that it depends largely on the supplier, none of their advertising is regulated, so you don't know for sure. You can also get peppercorns from other regions besides India and naturally the flavor varies - these are usually harder to find as they are not nearly as commercialized.
Enjoy the #1 spice.