My sons got me started on Doctor Who with the 2005 series restart. It is all available on Netflix instant, seasons one through five, or through sidereel.com (where you can watch 72 minutes at a time for free, and then you have to take an hour break before you can watch your next 72 minutes free, I don't know why, but that's their system.) We don't have TV service and are watching the current season the day after it airs on sidereel.
I have watched every new series episode, some more than once, each has it's merits, some are written by different people so there may be aspects you like more or less based on the writers style. The hardest part of watching is the gap time between seasons, but a post-season yearly Christmas special helps out! Overall the series is a lot of fun to watch. There are certainly some amazing episodes, like the weeping angels, and the visuals are very striking, and very funny, but scary too. The original series is historic, but really dated, moves slowly, and it is hard to watch unless you like the look of retro TV. The new series is action packed and very fast paced, that's why it holds up of multiple viewings of the episodes so well.
The Neil Gaiman episode is a perfect one to watch as a one off, because it shows all the existing themes and explores them, picking up and leaving off in exactly the same spot, but detailing an amazing facet no author touched on before.
I don't know of any other consistently brilliant current TV series.