If your new TV does not have audio video outputs you will need to buy a digital converter box for the VCR and then split your antenna to feed both the TV and converter box. The converter box connects to the VCR via the antenna or av inputs. If you use the antenna you set the VCR to channel 3 usually. There is a downside in that the VCR will not change the channels on the converter box so your programming flexibility is limited by this. Your VCR should connect to the AV inputs on the TV for playback.
Do you use a set-top box? Does it have analog outputs? Composite video? Component video? S-Video? Use those to connect to your vcr. If you have a coax output from your STB that may work when connected to your vcr, but only use this as a last resort. It's easy enough to test. The analog outputs (composite/component/s-video) will definitely work, assuming your vcr has these inputs -- it should have at least one of these.