I'm a whole generation older than you, so I started watching Tom Baker as the Doctor when I was a teen in the 70s. I have a friend the same age as me who lived in the UK as a child and watched the original series live as a small child. He remembers thinking it was craptastic even as a 5 year old (but he watched it anyway - not a lot of good options back then). I saw the shows from the 60s in my teens in the mid 70s and really couldn't get past the lack of production values. Revisiting them later on in my 30s, I still didn't find them really worth watching.
For me, the best Doctors were Tom Baker, David Tennant, Peter Davison, and Christopher Eccelston. I like Matt Smith, the current doctor as well.
I agree with your overall advice to the OP:
1. If you have have limited time, just start watching from the "reboot" of 2005.
2. If you have more time, start with Tom Baker, then continue on with his successors from the original series as long as your interest holds up.