All good advice here. Having wired 4 recording studios, a couple of garage/sheds, and countless indoor lighting fixtures, switch boxes, and outdoor outlets I can tell you that you don't want to use the typical orange extension cord in a burial installation. It will work, for a while, but then deteriorate and split and you're back to square one. Plus, you'd be looking at moisture/water issues with it. Do it right and be done with it. And you can do it all yourself with just a little basic know how.
Like others have said use a cable rated for direct burial. There are a variety of guages of direct burial cable available at HD or Lowes, both with and without conduit, depending on your circuit load requirements. if you just want to run a basic light fixture and maybe an outlet you should be okay without having to add an add'l circuit. Conduit is your best course for long term protection of the cable, but direct burial cable without conduit will also work as long as there's no chance of cutting thru it with digging later on. It rated for direct burial...so that's what it's for. Bury it deep enough to protect it from standing water or direct surface water seepage. 8-12 inches should do it. For a 150 foot run I'd rent a small trencher and make the job quick and easy and maybe put it down deeper.
I recently added a circuit and trenched a 155 foot run to power the submerged pump of a water well. The trencher was a lifesaver...well worth the $100 bucks for a half day rental...it dug the trench in about 20 minutes.
I would suggest the simplest hook up would be tapping into the outlet you had planned on using the extension cord with and hardwiring the connection, although you could get a right angle plug and attach it to the buried cable and just plug it in. On the shed end of the line you could also connect a dual or quad outlet and then just daisy chain up the wall/ceiling to a simple light fixture.
Yes, you're looking at a couple of hundred bucks - but it's worth it to do it right.