If you're skinny or on the skinny side, you will see aesthetic result within a few weeks, trust me. But I still recommend full body routines (and arm work as well), noticing your whole body change before your eyes is a tremendous confidence/self esteem boost, especially when others start noticing as well. You won't want to stop working out when that happens. I can tell you this because that's what I am doing currently, squat/deadlift/press/bench press along with some arm exercises, has worked out splendidly.
Eat big, lift heavy, enjoy the results. Don't be discouraged if you have to start out light, it's a work in progress thing. Starting strength in the OP is a great program.