Here's my take.
I started with Doctor Who watching the odd PBS episode when I was a kid and Tom Baker was on, thinking to myself "MAN this looks dumb." I was just a kid, then, and had grown up on Star Wars and the old BSG and whatnot. I went back to Doctor Who ages later, watching a few episodes from Tom Baker's Key To Time season (a loosely organized collection of otherwise unrelated stories strung together by a single plot thread that barely makes appearances in those stories).
I later got into the 2005 Series with Eccleston as the Doctor, and LOVED the show. Based on that, I figured "What if I gave the old stuff a chance again?" Went back, and watched the whole of The Key to Time (which I bought). I liked it, and so I decided to go ALL the way back and start from the beginning, watching in chronological order.
Make no mistake -- some of the early stuff is tough to get through sometimes, ESPECIALLY if your main exposure to the Doctor is via the new series. The new series is entertaining but moves at a breakneck pace by comparison with the old.
However, the old series really DOES have a lot to offer and to recommend it. Come on, man, you get to see the FIRST appearance of the Daleks! EVER! And it's a GOOD story.
That, for me, is the core of the older Who stuff. It's genuinely good stories, even if the execution fails from time to time. The characters are fun, and they become...I dunno...old friends for you after a while. You just get comfortable with them. Old Doctor Who to me is now a very comforting thing. Yes, even the Colin Baker years.
Also, with respect to the "family show" aspect, understand that a 1960s "family show" is not the pastel and happy-happy that you get in modern family shows. There's a quality to the acting that can be a bit broad at times, rather than the more modern "realist" approach. An Unearthly Child, however, is pretty real as the old series goes.
Also, side note: when you watch, I HIGHLY recommend turning on the info text on the DVDs. It's a LOT of fun and can be pretty informative, and the folks writing it don't take things 100% seriously all the time.