You can pretty much start anywhere. I first watched random episodes from Tennant's series on television when they first aired, and even if I missed an episode or two, I still followed them because they're mostly self-contained stories. It wasn't til a couple years later that I actually sat down and watched them in order from the start of Eccleston's run. (Mind you, if it's the last episode of a season or starts with a "previously on" segment, you're probably not going to fully understand it especially if you're slow like me.) And now that I'm finished with the reboot, I'm basically just watching random old episodes that sound interesting, or that my Da remembers. A Tom Baker episode one night, some William Hartnell the next. And it makes perfect sense.
As Neil Gaiman said,
“People are intimidated. They think that there’s 47 years worth of stuff they need to know before they can enjoy anything. And what you want to say is, No. Look, there is a blue box. It is bigger on the inside than it is on the outside. It can go anywhere in time and space and sometimes even where it’s meant to go. And when it turns up there’s a bloke in it called the Doctor, and there will be stuff wrong and he will do his best to sort it out and he will probably succeed, because he’s awesome. Now sit down, shut up, and watch “Blink.”