I wouldn't go with slimmer tires. It's only 6 commutes, and yes it makes you faster, but the wider, more off-road tires would make it more comfortable to hop curbs and road debris. Plus, I'm always worrying my tires/rims are getting shot with every bump I take...

Getting new cables should be a nice way to start, and not too expensive either. After this you'll naturally run into adjusting derailleurs and brakes. Brakes are the easy part, derailleurs are the 'hard' part. Check out youtube, there are some instructables around on how to adjust them.

As said (in a few posts before) grease is very important. It's easy to take a part the wheel axles and regrease the bearings in it. Just make sure you have proper bearing grease and don't lose any bearing balls.

Clean your bike each weak, or as you ride only 12 miles a day (equals 60 miles a week) every two weeks. Sand and other dirt can slow you down on your way as it increases rubbing alongside the brakes, or near the axles of the wheels.

A few things you might want, are a Topeak seatpost clipon luggage rack with a fitting fender, and a front fender. Getting your legs and your back wet after the rain isn't fun. And if you ever need to take something extra to work, you have a place to put it (which obviously goes for your clothing as well, since you do have to take it to work once in a while).

Not sure what Magna type you have, but did a quick Google search for it. You might want to go with clipless pedals in a few months, but it's not a necessity (yet). Make sure you have reflectors in your spokes, and nice and bright lights in the front and rear of your bike (lighting is very important when commuting). Maybe some bar ends for the extra hand positions, could decrease or prevent cramping/numb/tingly fingers while on the road.