My favorite strings to begin damaging combos are 123 and 113.
113, B3, J3, B23 and 123, B2, MB sky grapple, B3, J3, B23 are amazing combos coming from these strings. These strings also begin with Batman's light attack (one of the fastest attacks in the game.) So punishes turn into full combo if you use these strings. I know they can be ducked on the first hit, but most people don't react fast enough to poke you out of them. Both of these strings also end in either a low or overhead attack which can cause some confusion.
Other strings I use are B113, B112 and 223. 223 is excellent if you can land a J2 as you can go in to one of Batman's easiest combos ( J2, 223, b3, J3, B23.) It can be poked out of, but if you hit confirm the J2 you can avoid this.
B113 is a good low starter that I like to throw out when putting pressure after a knockdown. This works well if you have been using Batman's overhead option (I'll list in shortly.) You can go in to: B113, 123, B2, MB sky grapple, B3, J3, sparks for good damage out of B113. B112 is a good compliment to B113 as it is safe and hits overhead.
Now, you can also use B23 as it is an amazing string. It covers a good portion of the air space in front of Batman making it a decent anti air. It also hits overhead in the second part of the string. You can apply pressure with this to hit crouching opponents that refuse to guard high. It also puts Batman away from opponents if you wish to zone. In fact, I often cancel this string into batarangs. Another overhead option is to MB F3 on knocked down opponents. This way you can absorb a hit from a wake-up attack and get a full combo from pressure.
Now, to make your approaches safe you MUST learn to use his trait properly. The trait gives you the ability to lock the opponent down and "get in" safely. You can essentially have one bat up at all times if you manage them correctly. These bats allow you to release them and dash in from pressure to attempt a jump in or one of the other various strings. These bats will also cover the holes in Batman's strings. If you have a bat up and find yourself in a situation that say, 223 is being blocked, you can release the bats before the hole in the string and be entirely safe. In fact, I have sometimes released one bat, dashed in, called another bat and got to the opponent as the first bat was hitting to apply pressure and still have one out to cover holes in strings. Batman's trait is amazing for accomplishing anything he needs to accomplish.
