I've lived in San Antonio pretty much all my life and the only area of town I recommend is the North Central side. I lived on the NW side of town for about 15 yrs until it got so bad we had to move. I had the tires and rims stolen off my car in my OWN driveway. I now live in the Stone Oak area and LOVE it. I wouldn't live anywhere else in town! I love being minutes away from good restaurants and lots of shopping!
As for school districts...the Northside school district used to be the premier district, but you'll find that the NEISD is up and coming as well. The price range you've listed is about the median for the area, so you'll have no problem finding a house. If you're looking for a little bit older neighborhood that has an East Dallas-Lakewood feel, then look at Alamo Heights (78209). Alamo Heights is very nice, but you're probably going to get less house for your money.
Here are the main zip codes for the NC side of town: Stone Oak - 78258, 78259, 78260; Deerfield- 78248; Shavano- 78231; Gold Canyon and surrounding area- 78232.
My main advice to you is to steer well clear of the South, West, and East sides. You will not be safe there. My grandma lives on the west side and has an almost weekly attempted break-in. If she weren't so stubborn and her house worth so little, we'd have moved her out of there years ago. Also, do not look anywhere inside Loop 410, unless its Alamo Heights, Olmos Park, or Castle Hills.