I've actually been trying to watch all of the Classic Who because I'm very interested in the origins of the show, and it's been difficult, and even frustrating at times (The Web Planet... *shudder*) But I feel like it's worth it.
I'm still on episodes from the Hartnell era (the First Doctor) and even though it's the very earliest form of the show, and as many people have said, you don't need to watch it for continuity purposes, I still find it very worthwhil and enjoyable (most of the time).
However, I'm a huge HUGE fan of Doctor Who so I am most definitely biased. It really comes down to personal taste, and your tolerance for very "60's" acting and cheesy and sometimes downright awful fx. The pace and plotting threw me through a loop at first, but you get used to it.
Anyways, Hartnell is a great Doctor, even if he takes a while to grow on you, and you should warm up to the new companions pretty quickly. Ian Chesterton is one of my favorite companions for sure.
TL;DR
Yes, you should watch Classic Who. It's awesome and worth it.