I would say tailing. That cop has been tailing me for a few blocks sounds the most normal.
Tailgating is not quite what you're looking for—that implies following someone too close, and shadowing is for other contexts.
To further substantiate my answer (from the Oxford American Writer's Thesaurus)
informal the paparazzi tailed them: follow, shadow, stalk, trail, track, hunt, hound, dog, pursue, chase.
on someone's tail a police car stayed on his tail: CLOSE behind, following closely, (hard) on someone's heels.
From those examples, stalk adds a whole slew of other implications to it, trail is similar and sounds awkward to me, hunt, hound, and dog also add implications. Pursue is also a good one, but isn't quite slang, is it? And chase means to chase.
And, I have even more for you—if you're okay taking it beyond slang. In my personal experiences driving around with my friends, I've said and heard things like that guy's creeping and that guy's been on my ass for a few blocks now.
if the words suggested already aren't good enough for you, you have to make something up yourself. Whatever works with people you know, in your area. This guy on YouTube, I'd imagine, was doing just that.