If you happen to have a weighted digital piano, get ready for one of the coolest things about studying with me.
One of the first things I do with my students is teach them how to play realistic LH walking bass lines. My background as a Hammond Organist and training from some of the masters makes me much more skillful at this than your typical Jazz pianist.
You might wonder: What's the big deal? Why would a pianist need to get good at playing LH bass? Well, first of all, it sounds great, and think how great it is to be able to make your own play-a-longs on the spot, but those aren't the main reasons. The big thing is that Jazz *is* contrapuntal improv. You have to get really good at listening to the bass player, getting the bass player in your head, and playing lines against walks. Once you can do this, you're playing real Jazz.
Oh, here's some great news: walking bass is not so difficult. I actually wrote one of the books on the subject, and can get most of my students accompanying themselves in fairly short periods of time. -This may sound like a"Jazz organ" approach to learning piano, I assure you it isn't. Take pioneer Bebpper Lennie Tristano (one of the father's of modern Jazz education.) Bass counterpoint was a very important element of his teaching and performance alike. Furthermore, back in the day, that was how teachers got you started: They'd get you accompanying yourself playing bass figures in your left hand and you'd alternate practice between LH chords and LH bass against RH melodies and chords.
As for students who only have real piano (poor souls, with nothing more than a well tuned baby grand in their living room), all is not lost. Sometimes creating your own play-a-long track is simple as plugging a cheap midi keyboard into your computer and bopping a track out just the same. Then again, you can still practice the old fashioned way with roots and occasional walks in the left hand. (See ex. 2 above.)