The most spectacular sights around Aleppo are the "dead cities" and St. Simeon, both covered in other entries, but, if you have a couple of days in Aleppo and are not short on time, I recommend extending your excursion to St. Simeon to include a few of the other sights buried in the limestone mountains. You can easily see them all (including St. Simeon and the dead cities) on one day-long excursion. Public transportation in this area is minimal and most of the sites are beyond the reach of minibuses, so if you have rented a car (not a bad idea for touring Syria) then you will have no trouble. If you do not have a car, it is perfectly easy to arrange a tour through your hotel. It should cost around $20/person.
The first of the major sights, and the one closest to Aleppo, is the Byzantine church of Mushabbak. The church, while nowhere near the scale of St. Simeon, is still impressively preserved and provides a good introduction to what you can expect from the other sites. The church is a basic basilica built in the fifth century. Inside, the nave and the support columns give you the best idea of what the church looked like in its prime. Also, just outside the church is a small quarry (where the stones for the church came from) and a small tomb. There is also a small monastery complex.
If you are on a planned excursion, you will head from here to a dead city and St. Simeon, after which you can proceed on to the temple complex of Ain Dara. Ain Dara is a Hittite temple from around 10,000 B.C. While the ruins themselves are not terribly impressive, they are still worth a look. The lion statues outside of the tomb give you a good idea of Hittite sculpting, and the large "footprints" inside the temple are unique. If you have never seen a Hittite temple before, Ain Dara will impress, and, in addition to great views, the most impressive aspect of the temple is merely its age.
After Ain Dara, you will head a good 30km farther north, probably stopping for lunch on the way. Beware that you will be expected to pay for your driver’s lunch if you stop. Also, you should know that the driver might just decide to bring his son along, like he did with us, and you will have to pay for him too. Admittedly I was a bit annoyed, but the 10-year-old boy was too cute, and it’s not like it was expensive anyway. The stretch of road from Ain Dara to the ruins of Cyrrhus is the most impressive part of the drive. You will wind around the mountains on narrow roads and through villages. Eventually at one point you will arrive at two Roman bridges, both immaculately preserved and still in use. Your driver will allow you to take some pictures before you head over the steep arches yourself. A few kilometers later, you will reach the ruins of Cyrrhus, an old Roman town. Not much remains except for a Roman tower and a crumbled theater, but the real reason is to head out there is for the drive and the views at the site, which is right on the Turkish border.
These sites may not compare to the grandeur of St. Simeon, but it is still a worthy day trip from Aleppo and one worth taking. The scenery is unbeatable.