We switched to DirecTV and I really enjoy the service. However, I tried to use my VCR a while ago to tape a show off TV and it did not record the program. In order to tape programs off TV now, will I have to upgrade to a DVR?

should be hooked up as follows. Back of direct TV reciever/ tv out to antenna in on the back of the VCR. On the VCR out to TV to your TV.
Always have your VCR on The same channel your watch your Sat. That will be either channel 3 or 4. Alway record that channel on your vcr/ chose the programm through the Sat. receiver.

In other words YES!!!!!!!!

I also have DirecTV, and I have been recording on the VCR for years with it. You have to remember that you are recording off the feed from the satellite dish, so make certain what channel the satellite is tuned to (and the satellite must be turned on, even if the TV is off). I set my system up in line so that the satellite feeds to the VCR, which then feeds to the TV, with never a day's trouble.